Different faces of generalized holographic dark energy

In the formalism of generalized holographic dark energy (HDE), the holographic cut-off is generalized to depend upon $L_\mathrm{IR} = L_\mathrm{IR} \left( L_\mathrm{p}, \dot L_\mathrm{p}, \ddot L_\mathrm{p}, \cdots, L_\mathrm{f}, \dot L_\mathrm{f}, \cdots, a\right)$ with $L_\mathrm{p}$ and $L_\mathrm{f}$ are the particle horizon and the future horizon, respectively (moreover $a$ is the scale factor of the universe). Based on such formalism, in the present paper, we show that a wide class of dark energy (DE) models can be regarded as different candidates of the generalized HDE family, with respective cut-offs. This can be thought as a symmetry between the generalized HDE and different DE models. In this regard, we consider several entropic dark energy models - like Tsallis entropic DE, the R\'{e}nyi entropic DE, and the Sharma-Mittal entropic DE - and showed that they are indeed equivalent with the generalized HDE. Such equivalence between the entropic DE and the generalized HDE is extended to the scenario where the respective exponents of the entropy functions are allowed to vary with the expansion of the universe. Besides the entropic DE models, the correspondence with the generalized HDE is also established for the Quintessence and for the Ricci DE models. In all the above cases, the effective equation of state (EoS) parameter corresponds to the holographic energy density are determined, by which the equivalence of various DE models with the respective generalized HDE models are further confirmed. The equivalent holographic cut-offs are determined by two ways: (1) in terms of the particle horizon and its derivatives, (2) in terms of the future horizon horizon and its derivatives.

be carried for two cases: (1) where the respective exponents of the entropy functions are treated as constant, while in the second case, (2) the exponents are allowed to vary with cosmic time, in particular for the latter case, the exponents are considered to depend on the evolving Hubble parameter of the universe. Besides the entropic DE models, the Quintessence [71][72][73][74] and the Ricci-DE [23,29,76,77] models will also take part in the present analysis. In case of Quintessence model, a non-minimally coupled scalar field with an exponential potential provides the dark energy density, while in the Ricci-DE scenario, the space-time Ricci curvature serves the dark energy density. Both the Quintessence and the Ricci-DE models turn out to be viable with respect to various dark energy observations. Interestingly, we will show that all such entropic DE, Quintessence and the Ricci-DE models are indeed equivalent with the generalized HDE, with suitable forms of the corresponding cut-offs.

II. THERMODYNAMICS OF SPACE-TIME AND APPLICATION TO COSMOLOGY
After the thermodynamical properties of the black hole were clarified [78,79] and it has been claimed that the entropy of the black hole is proportional to the area A of the horizon which is called as the Bekenstein-Hawking entropy, r H is the horizon radius, and we work in units where = k B = c = 1, there have been long and active studies where the connection between the gravity and the thermodynamics could be clarified [80][81][82]. In the studies, we have found that the FRW equations can be also regarded as the first law of thermodynamics when we consider the Bekenstein-Hawking entropy by using the cosmological apparent horizon [83][84][85] as a realization of the thermodynamics of space-time [80].
In case that, however, there are long range forces like the electro-magnetic one or gravitational one, we know that the systems are non-additive systems and the standard Boltzmann-Gibbs additive entropy should not be applied and we should generalize the entropy to the non-extensive Tsallis entropy [49][50][51] and recently there are several attempts in this regard (see [52][53][54][55][56][57][58][59][60][61][62]). If we apply the Tsallis entropy to the black hole, instead of the Bekenstein-Hawking entropy, one finds [52], In the above expression, A 0 is a constant and δ is the new parameter that quantifies the non-extensivity. Then if we apply the Tsallis entropy by using the apparent horizon to the cosmology, the FRW equations should be modified and the modification can be regarded as the contribution from the dark energy. In information theory, the Rényi entropy is often used as the measure of the entanglement. If we apply the Rényi entropy to the black hole, one finds [63][64][65][66][67] S R = A 0 Gδ ln 1 + δ 4 The Rényi entropy has been also used to explain the dark energy [67].
Here it may be mentioned that both the Tsallis and Rényi entropy expressions belong from one-parametric entropy family; there is also a two-parametric generalized entropy which is called the Sharma-Mittal entropy (S SM ) and is written as [67,69,70], where A 0 is a constant, α and δ are two independent parameters. Some cosmic applications of the Sharma-Mittal entropy can be found in [69] where the Hubble horizon plays the role of cut-off and moreover no mutual interaction between the cosmos components has been taken into account. Above considerations of different entropies eventually lead to different scenarios of holographic dark energy, which will be discussed in the following two sections.

III. DARK ENERGY CORRESPONDING TO TSALLIS, RÉNYI, AND SHARMA-MITTAL ENTROPIES
We assume the Friedmann-Lemaître-Robertson-Walker (FLRW) space-time with flat spacial part, whose metric is given by Here a(t) is called as a scale factor. If we define the Hubble rate H by H =ȧ a , the radius r H of the cosmological horizon is given by Then the entropy in the region inside the cosmological horizon could be given by the Bekenstein-Hawking relation [82] in (1). On the other hand, the flux of the energy E or the increase of the heat Q in the region is given by where we use the conservation law: 0 =ρ + 3H (ρ + p). Then by using the Hawking temperature [83] and the first law of thermodynamics T dS = dQ, one obtainsḢ = −4πG (ρ + p) and by integrating the expression , one obtains the first FLRW equation, Here the cosmological constant Λ appears as a constant of the integration. Instead of the Bekenstein-Hawking entropy (1), we may use the non-extensive, the Tsallis entropy [49][50][51]54] in (2). Then by applying the first law of thermodynamics to the system, instead ofḢ = −4πG (ρ + p), one gets [55] δ on integrating which, one gets, Here a constant H 1 is defined by A 0 ≡ 4π . Then if we define the energy density ρ T and the pressure p T by respectively. It is evident that ρ T depends on the quadratic power of the Hubble parameter and thus is symmetric with respect to the Hubble parameter. With the above forms of ρ T , p T , Eqs. (10) and (11) can be expressed aṡ respectively. Therefore ρ T and p T represent the energy density and pressure correspond to Tsallis entropy. Consequently the respective equation of state (EoS) parameter for the Tsallis entropy is given by, It may be checked that the above expression of ω T leads to the conservation equation for the Tsallis entropic energy density, i.eρ Here it deserves mentioning that the authors of [55] showed that the ω T in Eq.(15) leads to a viable dark energy epoch of our present universe, where the matter sector is considered to be dust. Moreover, the analysis is also extended to the case where the radiation energy density is present too. In particular, due to the Tsallis entropic energy density, the universe exhibits the usual thermal history, with the sequence of matter and dark-energy eras and the onset of acceleration occurs at around z ≈ 0.5 which is in agreement with observations [55].
In regard to the Rényi entropy (3), the first law of thermodynamics gives, from which, we obtain Here the cosmological constant Λ appears as a constant of the integration again. At this stage we may define the corresponding energy density and the pressure in the following form Similar to the Tsallis entropic case, the Rényi entropic energy density (i.e ρ R ) seems to be symmetric in respect to the Hubble parameter. Due to the above expressions of ρ R and p R , Eqs. (17) and (18) become similar to the usual Friedmann equations where the total energy density and total pressure are given by ρ eff = ρ + ρ R and p eff = p + p R . Consequently, the EoS parameter corresponds to the Rényi entropy comes with the following form, It may be mentioned that the above expression of ω R obeys the conservation equation for the Rényi entropic energy density. As showed in [67,68], the Rényi entropic energy density (ρ R ) and the pressure (p R ) can provide suitable description for the current accelerated universe and thus leads to a dark energy model. In case of the Sharma-Mittal entropy, the first law of thermodynamics leads to the following evolution of the cosmic Hubble parameter, integrating which, we obtain, where Λ is the constant of integration, 2 F 1 is the hypergeometric function, and to get the above expression, we use the conservation equation of the matter components. Moreover, the constant H 1 is related to A 0 by A 0 = 4π . Now if we define an energy density (ρ SM ) and a pressure (p SM ) like, respectively, then Eqs. (22) and (23) can be equivalently expressed as, Thus we may argue that ρ SM and p SM are the energy density and the pressure coming from the cosmological description of the Sharma-Mittal entropy. Furthermore ρ SM and p SM are connected by the respective EoS, as given by where we use Eqs. (24) and (25). The above form of ω SM immediately confirms the conservation equation for the Sharma-Mittal entropic energy density. Furthermore, as established in [67,69], the Sharma-Mittal entropic energy density leads to a late time acceleration epoch of our universe. In [69], the universe in considered to be filled by a pressureless component and Sharma-Mittal entropic energy density, which do not have any mutual interaction, and as a result the present deceleration parameter is found to be consistent with the present observation.
Before closing this section, here we would like to mention that the presence of entropic energy densities indeed modify the FLRW equations. Such modifications can also be encapsulated in the respective entropy functions. In particular, when we consider the Bekenstein-Hawking entropy in the context of cosmology, one gets the usual FLRW equations and thus we have the expression like dS dH = −2π/ GH 3 which leads to S = S(H). However for the modified entropy cases, the relation of S = S(H) will become different compared to the Bekenstein-Hawking case. In particular, for the Tsallis entropy case, the first FLRW Eq.(11) leads to the following expression: which along with the first law of thermodynamics (in Eq. (7)) yield the rspective entropy in terms of the Hubble parameter as, on integrating which, one gets S T = S T (H). It is evident that for δ = 1, the above expression becomes similar to that of the Bekenstein-Hawking case. Thereby the modification of the Tsallis entropy compared to the Bekenstein-hawking case is clearly demonstrated by the expression of S T = S T (H). By similar procedure, we can obtain the Rényi entropy and the Sharma-Mittal entropy functions in terms of the Hubble parameter as, Clearly Eq.(30) depicts that for δ = 1, one gets dSR dH = −2π/ GH 3 , while Eq.(31) reveals that the situation α = δ leads to dSSM dH = −2π/ GH 3 , i.e they become similar with that of the Bekenstein-Hawking entropy function for the aforesaid conditions respectively. Here we would like to mention that Eqs. (29), (30) and (31) remains symmetric under the transformation H → −H.

IV. GENERALIZED HOLOGRAPHIC ENERGY
In the holographic principle, the holographic energy density is proportional to the inverse squared infrared cutoff L IR , which could be related with the causality given by the cosmological horizon, Here κ 2 = 8πG is the gravitational constant and c is a free parameter. The infrared cutoff L IR is usually assumed to be the particle horizon L p or the future event horizon L f , which are given as, Differentiating both sides of the above expressions lead to the Hubble parameter in terms of L p ,L p or in terms of L f , L f as, In [9], a general form of the cutoff was proposed, Actually, the other dependency of L IR , particularly on the Hubble parameter, Ricci scalar and their derivatives, can be transformed to either L p and their derivatives or L f and their derivatives via Eq.(34). The above cutoff could be chosen to be equivalent to a general covariant gravity model, We will use the above expressions frequently in the following sections. With the help of the generalized cut-off, we aim to show that the Tsallis, Rényi and Sharma-Mittal entropic dark energy may belong from the generalized dark energy family where the holographic cut-offs are expressed in terms of the particle horizon and its derivatives or in terms of the future horizon and its derivatives. The comparison of Eqs. (12) and (32) lead to the argument that the Tsallis entropic dark energy belongs from the generalized holographic dark energy family, where the corresponding infrared cutoff L T is given by, in terms of L p and its derivatives. To get the above expression, we use Eq. (34). Moreover, L T in terms of the future horizon and its derivatives comes by the following way, Here we would like to determine the EoS parameter of the holographic energy density corresponds to the cut-off L T , in particular of ρ hol = 3c 2 / κ 2 L 2 T . In this regard, the conservation equation of ρ hol immediately yields the respective EoS parameter (symbolized by Ω where L T is obtained in Eq.(37) (or Eq. (38)) and the superscript 'T' in the above expression denotes the EoS parameter corresponds to the holographic cut-off L T . Due to Eq.(34), the above form of Ω (T ) hol seems to be equivalent to the EoS of the Tsallis entropic energy density presented in Eq.(15), i.e Ω (T ) hol ≡ ω T . Such equivalence, along with the fact that the Tsallis entropic energy density provides a viable dark energy model, lead to the argument that the holographic energy density coming from the cut-off L T is also able to produce a viable dark energy epoch at our current universe.
Similarly, by comparing (19) and (32), the infrared cutoff L R corresponding to the Rényi entropy is given by where, once again, we use Eq. (34). The first expression of Eq. (40) gives the L R in terms of L p and its derivatives, while the second one represents the same in terms of L f and its derivatives. Once again, the conservation equation of the holographic energy density ρ hol = 3c 2 / κ 2 L 2 R leads to the corresponding EoS parameter (Ω where L R is given in Eq. (40). It can be easily checked that the Ω hol ≡ ω R . Thereby, since the Rényi entropic energy density suitably describes the current acceleration of our universe, we may argue that the holographic energy density coming from L R is able to produce the late time cosmic acceleration.
Finally Eqs. (24) and (32) clearly argue that the Sharma-Mittal entropic dark energy can also be thought as one of the candidates of the generalized dark energy family, where the corresponding cut-off (L SM ) is given by, in terms of the particle horizon and its derivatives. Similarly, the L SM in terms of the future horizon and its derivatives is given by, Furthermore using the conservation relation of ρ hol = 3c 2 / κ 2 L 2 SM , we determine the EoS parameter (Ω hol ) corresponds to the holographic energy density coming from the cut-off L SM as, where L SM is given in Eq.(42) (or in Eq. (43)). In effect of Eq.(34), it is evident that the above form of Ω = ω SM . Due to this equivalence, we may argue that the holographic energy density ρ hol = 3c 2 / κ 2 L 2 SM can produce the late time acceleration of our universe.
Therefore the dark energy models coming from the Tsallis entropy, the Rényi entropy and the Sharma-Mittal entropy can be thought as different candidates of the generalized holographic dark energy family, where the respective infrared cutoffs are given by Eq. (37) to Eq. (43) respectively. Thereby such holographic cut-offs establish a symmetry between generalized HDE and the respective entropic DE model(s).

V. EXTENDED CASES OF ENTROPIC DARK ENERGY MODELS
In this section we consider the models extended as in [86], where the non-extensive exponent δ of the Tsallis entropy (2) or the Rényi entropy (3) depends on the energy scale and shows a running behavior [87]. In [87], it has been claimed that such behaviors may appear because the entropy corresponds to physical degrees of freedom and the degrees of freedom depend on the scale as implied by the renormalization of a quantum theory. In case of gravity, if the space-time fluctuates at high energy scales, the degrees of freedom may increase. On the other hand, if gravity becomes a topological theory, the degrees of freedom may decrease.
In cosmology, if we assume that the energy scale could be given by the Hubble scale H, δ in (2)  H 2 by introducing a parameter H 1 whose dimension is identical with H. Then in case of the Tsallis entropy (2), instead of (11), one obtains the following generalized first FLRW equation (see the appendix for the detailed derivation), and we may define the effective energy density ρ T by Thereby the presence of a varying exponent in the Tsallis entropy modifies the FLRW equations, which may have considerable impacts in the universe's evolution, both at high and low energy scales. Clearly, in order to determine the explicit expression of ρ T from the above equation, one needs a functional form of δ(x). Some of our authors proposed a suitable form of δ(x) in [87], which allows to analytically perform the integration in Eq. (46) and also leads to an unified scenario of early inflation with late time acceleration. Actually, the form of δ(x) is chosen in such a way that at high and low energy scales it acquires values away from the standard value 1, while at intermediate scales it comes close to unity [87].
In case of the Rényi entropy (3), the first law of thermodynamics gives (see the appendix for the detailed derivation), where δ ′ (x) = dδ dx . Then we may define the effective energy density ρ R corresponding to Rényi entropy by The terms containing δ ′ (x) in the above expression of ρ R arise due to the varying exponent δ = δ(x). Such terms may play an important role during the early as well as in the late cosmic evolution of the universe. In particular, we expect that at high and low energy scales, the δ(x) should deviate from the standard value unity and thus have a significant role in driving the inflation or the late dark energy epoch. However the modified cosmology from the Rényi entropy with a varying exponent has not been extensively studied in various earlier literatures. Thereby it will be an interesting avenue to study the possible effects of ρ R in Eq. (48) in the context of inflation, late time acceleration or even in the bouncing scenario. However these are out of the scope from the present work and thus we expect to study it in a future work. Coming back to the Sharma-Mittal entropy, there are two independent parameters (α and δ) and in the extended scenario, we take α = α(x) and δ to be constant. However, in the extended case of the Sharma-Mittal entropy, one may choose both the parameters being dependent on x = H 2 1 /H 2 , i.e., α = α(x) and δ = δ(x). For simplicity, here we stick to the aforementioned consideration, i.e., α = α(x) and δ = constant. Such consideration leads to the FLRW equation as (see the appendix for the detailed derivation), with, Consequently, the effective energy density corresponds to the Sharma-Mittal entropy comes by Eqs. (46), (48), and (51) immediately tell that the infrared cut-off L T , L R and L SM corresponding to the extended version of the Tsallis entropy, the Rényi entropy, and the Sharma-Mittal entropy are given by, and respectively, with f (x) is shown in Eq. (50). Therefore even for the extended case, the holographic energies coming from the Tsallis entropy, the Rényi entropy and the Sharma-Mittal entropy can be expressed by the general infrared cutoff in [9]. Interestingly, the corresponding cut-offs are determined in terms of the particle horizon and its derivative or in terms of the future horizon and its derivative.

VI. SOME OTHER DE MODELS AND THEIR EQUIVALENCE WITH GENERALIZED HDE
Besides the Tsallis, Rényi and Sharma-Mittal entropic DE scenario, some other DE models, in particular the Quintessence [71][72][73][74] and the Ricci-DE models [23,29,76], will take part in the present analysis. Their equivalence with the generalized HDE and the corresponding cut-offs are discussed in the following two subsections respectively.

A. Quintessence dark energy
The present observation indicates that the equation of state parameter at the present universe is close to ω ≃ −1, however, it says a little about the time evolution of ω, and thus we can broaden our situation and consider a dark energy model where the equation of state changes with time. Such kind of dark energy models are the scalar field dark energy models where the dynamics of the scalar field over the FLRW space-time leads to a evolving EoS of the universe. So far, a wide amount of scalar field dark energy models have been proposed, these include Quintessence, Phantoms, K-essence, Tachyon, Dilatonic dark energy etc.
In this section, we consider the Quintessence dark energy (QDE) model and aim to show that QDE is equivalent to the generalized holographic dark energy model where L IR = L IR L p ,L p ,L p , L f ,L f ,L f . The QDE action is given by, where φ is the Quintessence scalar field and V (φ) is its potential. The presence of the potential is important in the dark energy context, otherwise the energy density and pressure of the scalar field become equal, which in turn leads to a decelerating expansion of the universe. In particular, with V (φ) = 0, the FLRW scale factor of the universe evolves as a(t) ∼ t 2/3 and thus the scalar field model without potential is not compatible with dark energy observations. The Quintessence potential has the following form [71][72][73][74], with V 0 and p are constants. The Quintessence model with the above exponential potential has been extensively studied in [74] where it was shown that the potential of Eq.(56) leads to a viable dark energy model in respect to SNIa, BAO and H(z) observations. However the most stringent constraints on the dark energy EoS parameter (ω Q ) comes from the BAO observations, in particular −1 < ω Q < −0.85 [74]. The FLRW equations correspond to the action (55) are, where, due to the homogeneity, the scalar field is considered to be the function of time only. The first FLRW equation immediately leads to the Quintessence energy density as, where in the second line, we useḢ = −4πGφ 2 . Eq.(58) evidents that the Quintessence energy density is not symmetric with respect to the Hubble parameter, unlike to the case of entropic dark energy models (that we considered earlier) where the entropic energy density proves to be symmetric in respect to the Hubble parameter. The exponential form of the Quintessence potential (see Eq. (56)) allows the following solutions of the Hubble parameter and the scalar field as, respectively. Here t 0 being a fudicial time and V 0 , p, and t 0 are related by the following constraint equation, Furthermore the evolution of the Hubble parameter clearly indicates that in order to get an accelerating expansion of the universe, the parameter p is constrained to be p > 1. By using Eqs. (56) and (59), we can express the Quintessence potential in terms of the Hubble parameter as follows, Plugging back the above expression into Eq. (58), we get ρ Q in terms of H andḢ as, Furthermore the pressure in the present context is given by, which, along with Eq.(62) immediately leads to the corresponding EoS parameter as, Having set the stage, now we are in a position to show the equivalence between QDE and generalized holographic dark energy model. The comparison of Eqs. (62) and (32) immediately lead to the equivalent holographic cut-off (L Q ) corresponds to the QDE as follows, Thereby the QDE can be equivalently mapped to the generalized holographic dark energy model where the cut-off is the function of L p ,L p ,L p or the function of L f ,L f ,L f . Furthermore, the EoS parameter (Ω (Q) hol ) corresponds to the hologrphic cut-off L Q is given by, where L Q is shown above. Clearly, in accordance of Eq. (34), Ω hol becomes equivalent to the ω Q of Eq. (64). Such equivalence leads to the fact that similar to the Quintessence energy density, the holographic energy density coming from the cut-off L Q also provides a good dark energy model of our universe.

B. Ricci dark energy
In this section, we intend to establish that the Ricci dark energy (RDE) model has a direct equivalence to the generalized holographic dark energy model. The RDE model [23,29,76] catches a special attention as the dark energy density in this context has a geometric origin, in particular the dark energy density is given by, with R being the space-time Ricci scalar and α is a model parameter. The above expression of ρ RD along with its conservation equation lead to the corresponding EoS parameter as, where we use the explicit form of ρ RD and z = a −1 −1 is known as the red-shift factor. It is evident that the parameter α actually controls the evolution of the ω RD and hence the universe's evolution. In particular, it has been showed in [29] that for 1/2 < α < 1, the RDE has EoS −1 < ω RD < −1/3 and for the case α < 1/2, the RDE start from quintessence-like and evolves to phantom-like. In regard to the observational compatibility of RDE, the parameter α is constrained by α = 0.394 +0.152 −0.106 from SNIa only (1σ), however a joint analysis of the SNIa+CMB+BAO observations gives a much tighter constraint on α as α = 0.359 +0.024 −0.025 [29]. Eqs. (67) and (32) indicate that the RDE has a direct equivalence to the generalized holographic dark energy model, where the corresponding the cut-off (L RD ) can be expressed as, in terms of L p ,L p andL p . Similarly, the L RD in terms of future horizon and its derivatives is given by, Such holographic cut-offs establish a symmetry between the RDE and generalized HDE. Furthermore a modified form of RDE has been proposed in [77], where the dark energy density comes with the following form, with α and β are two parameters. The comparison of the above equation with Eq. (32) immediately leads to the equivalence holographic cut-off (in terms of L p and its derivatives or in terms of L f and its derivatives) corresponds to the modified RDE as, where we use Eq. (34). The EoS parameter (Ω (RD) hol ) corresponds to the holographic energy density ρ hol = 3c 2 / κ 2 L 2 RD arises from its conservation relation, in particular we get which, due to Eq.(34), evidents to be equivalent with the ω RD of Eq. (68). Thus similar to the RDE, the holographic energy density having the cut-off L RD proves to be a viable dark energy model of the universe in regard to the SNIa+CMB+BAO observations. Therefore the RDE and the modified RDE may be regarded as certain candidates of the generalized holographic dark energy family, with the respective holographic cut-offs given by Eqs. (69), (70), and (72), respectively.
Before concluding we consider the scale invariant cosmological field equations [88] and investigate its holographic correspondence. The said field equations are given by [88], where ρ and p represent the energy density and pressure of the matter components. Moreover λ parametrizes the scale invariance, which varies with the expansion of the universe, i.e λ = λ(a(t)). Here it may be mentioned that the authors of [88] proposed an inflationary scenario in the context of scale invariance cosmology, in which case the matter components are provided by a slow rolling scalar field, in particular ρ = 1 2 C Ψ 2 + U (Ψ) and p = 1 2 C Ψ 2 − U (Ψ) , where Ψ is a scalar field, U (Ψ) being its potential and C is a constant (for more information about C, see [88]). Clearly the field Eqs. (74) and (75) can be equivalently mapped to the holographic cosmological scenario, where the holographic cut-off and the corresponding EoS parameter are given by, or , and Ω (SI) Eqs. (76) and (77) represent the L SI in terms of L p (and its derivative) and L f (and its derivative) respectively. The EoS parameter in Eq.(78) satisfies the conservation relation of the holographic energy density ρ hol = 3c 2 / κ 2 L 2 SI . As a whole, the scale invariant cosmological model (described by Eqs. (74) and (75)) has a holographic correspondence, with the cut-off being given in Eq. (78).
At this stage it deserves mentioning that as far as we could see on many examples of modified gravity, scalar-tensor theory or gravity with fluids the corresponding FLRW equations can be always mapped to holographic cosmology with specific IR cut-off. However, of course the physical nature of such cut-off remains to be obscure.

VII. CONCLUSION
Dark energy (DE) is one of the most puzzled issues in modern cosmology. In particular, DE may even be an issue of quantum gravity. In this regard, the holographic principle, one of the important cornerstones of quantum gravity, plays an important role in describing the dark energy of our universe. Based on the holographic principle and on the dimensional analysis, the theory of holographic dark energy (HDE) has been formulated, where the dark energy density is proportional to the inverse squared if the infrared cut-off. The holographic cut-off is usually considered to be same as the particle horizon or the future horizon. It may be stressed that instead of adding a term into the Lagrangian, the HDE is based on the holographic principle and on the dimensional analysis and this makes the HDE significantly different than the other theory of DE. In [9], a generalized HDE has been proposed where the cut-off (L IR ) is generalized to be a function of particle horizon (L p ) and its derivatives of any order or a function of future horizon (L f ) and its derivatives of any order, in particular L IR = L IR L p ,L p ,L p , · · · , L f ,L f , · · · , a . Evidently, with such generalized form of the L IR , the phenomenology of the generalized HDE becomes more richer.
Based on the formalism of the generalized HDE, we showed that a wide class of dark energy models can be regarded as different candidates of the generalized holographic dark energy family with respective cut-offs. In this regard, we first considered several entropic DE models, in particular the Tsallis entropic DE, the Rényi entropic DE, and the Sharma-Mittal entropic DE, and showed that they are indeed equivalent to the generalized HDE model, where the corresponding cut-offs are determined in terms of L p ,L p or in terms of L f ,L f , respectively. Such equivalence between the entropic DE and the generalized HDE are established for two cases: (1) in the first case, the exponents of the respective entropy functions are regarded to be constant, while in the second case (2) the exponents vary with cosmic time, particularly the exponents are considered to depend on the evolving Hubble parameter. Here it may be mentioned that for the entropic DE models, the equivalent holographic cut-offs depend up-to the f irst derivative of L p or L f . Besides such entropic DE models, some other DE models like -(1) the Quintessence model where a minimally coupled scalar field with an exponential potential serves the dark energy density and (2) the Ricci DE where the space-time curvature provides the dark energy density -are also proved to be equivalent with the generalized HDE. The equivalent holographic cut-off corresponds to the Quintessence as well as to the Ricci DE model depends on either L p ,L p ,L p or L f ,L f ,L f . It may be noted that for both the Quintessence and Ricci DE models, the equivalent cut-offs depend up-to the second derivative of L p or L f , unlike to that of the entropic DE models where, as mentioned earlier, the corresponding L IR depends at most on the f irst derivative of the L p or L f respectively. Finally it deserves mentioning that in all the cases, we determine the effective EoS parameter for the DE models and the corresponding generalized HDE models, where the EoS parameter are represented by ω i and Ω (i) hol respectively (with i denotes the various cases we considered). As a result, we found that ω i ≡ Ω (i) hol , which further confirms the equivalence between various DE models and the respective generalized HDE models. This indicates a symmetry between the generalized HDE and different DE models.
In summary, a wide class of dark energy models including the entropic DE models are found to be equivalent with the generalized HDE, with the corresponding cut-offs being determined in terms of the particle horizon and its derivatives or in terms of the future horizon and its derivatives. However, the understanding for the choice of fundamental viable cut-off still remains to be a debatable topic. The comparison of such cut-offs for realistic description of the universe evolution may help in better understanding of holographic principle. Furthermore, it is interesting to note that recently holographic inflation [38] was proposed with above generalized holographic cut-off. Our considerations indicate that similar equivalence may be established between different inflationary theories and holographic inflationary model with generalized cut-off. The Tsallis entropy with varying exponent, in particular δ = δ(x) where x = H 2 1 /H 2 , is given by, Therefore, which is written in Eq.(45).

Derivation of Eq.(47)
The Rényi entropy with varying δ = δ(x) is given by, Therefore, Plugging the above expression into Eq.(7), one gets the evolution ofḢ in the present case as,