Kaniadakis Holographic Dark Energy

• KHDE is a dark energy model where standard Bekenstein–Hawking entropy is replaced by Kaniadakis entropy, leading to modified holographic density or Friedmann equations.
• Different formulations emerge based on the chosen infrared cutoff and gravitational framework, producing dynamics that range from quintessence-like to phantom-like behaviors.
• Observational analyses generally constrain the Kaniadakis deformation to be small, with some implementations aligning closely with ΛCDM while others help address tensions like the Hubble constant discrepancy.

Kaniadakis holographic dark energy (KHDE) denotes a class of dark-energy models in which the Bekenstein–Hawking entropy of a cosmological horizon is replaced by the Kaniadakis entropy, $S_{K}=(1/K)\sinh(KS_{BH})$, and the holographic bound or the gravity–thermodynamics conjecture is then used to derive either a modified dark-energy density or modified Friedmann equations. In the current literature, KHDE is not a single unique construction: implementations differ by the infrared cutoff—future event horizon, Hubble horizon, apparent horizon, or generalized future event horizon—and by the gravitational framework—flat or non-flat FRW, Brans–Dicke cosmology, Kaniadakis-modified horizon thermodynamics, and 4D Einstein–Gauss–Bonnet gravity (Drepanou et al., 2021, Lymperis et al., 2021).

1. Entropic basis and holographic prescription

The Kaniadakis deformation originated as a one-parameter generalization of Boltzmann–Gibbs entropy. In the statistical formulation used in the KHDE literature, one writes

$$S_{K}=-\sum_{i=1}^{W}\frac{P_i^{1+K}-P_i^{1-K}}{2K}, \qquad -1<K<1,$$

which reduces to the standard Boltzmann–Gibbs form as $K\to0$. Under the equiprobability assumption $P_i=1/W$ and the identification $W=\exp(S_{BH})$, this becomes the horizon expression

$$S_{K}=\frac{1}{K}\sinh(KS_{BH}), \qquad S_{BH}=\frac{A}{4G}.$$

For small deformation, the expansion

$S_{K}=S_{BH}+\frac{K^2}{6}S_{BH}^3+\mathcal O(K^4)$

shows that the leading correction is cubic in $S_{BH}$ (Drepanou et al., 2021).

KHDE then follows by inserting $S_K$ into the holographic relation $\rho_{DE}L^4\lesssim S$, with $$S_{K}=-\sum_{i=1}^{W}\frac{P_i^{1+K}-P_i^{1-K}}{2K}, \qquad -1<K<1,$$0 an infrared cutoff, or by inserting $$S_{K}=-\sum_{i=1}^{W}\frac{P_i^{1+K}-P_i^{1-K}}{2K}, \qquad -1<K<1,$$1 into the Clausius relation on the apparent horizon. These two routes are structurally different. The first yields a modified holographic density, whereas the second yields modified Friedmann equations whose extra terms can be recast as an effective dark-energy fluid (Hernández-Almada et al., 2021).

A common source of confusion is that these constructions are sometimes grouped under the same label without distinguishing their dynamical content. The event-horizon version, the apparent-horizon first-law version, and the Hubble-cutoff $$S_{K}=-\sum_{i=1}^{W}\frac{P_i^{1+K}-P_i^{1-K}}{2K}, \qquad -1<K<1,$$2 approximation are mathematically inequivalent, even though all are entropically Kaniadakis-based.

2. Principal formulations of KHDE

In the future-event-horizon formulation, the infrared cutoff is

$$S_{K}=-\sum_{i=1}^{W}\frac{P_i^{1+K}-P_i^{1-K}}{2K}, \qquad -1<K<1,$$3

and the KHDE density takes the form

$$S_{K}=-\sum_{i=1}^{W}\frac{P_i^{1+K}-P_i^{1-K}}{2K}, \qquad -1<K<1,$$4

The first term is the standard holographic contribution; the second is the Kaniadakis correction. In a flat FRW background this leads to a closed evolution equation for $$S_{K}=-\sum_{i=1}^{W}\frac{P_i^{1+K}-P_i^{1-K}}{2K}, \qquad -1<K<1,$$5, from which $$S_{K}=-\sum_{i=1}^{W}\frac{P_i^{1+K}-P_i^{1-K}}{2K}, \qquad -1<K<1,$$6, $$S_{K}=-\sum_{i=1}^{W}\frac{P_i^{1+K}-P_i^{1-K}}{2K}, \qquad -1<K<1,$$7, and $$S_{K}=-\sum_{i=1}^{W}\frac{P_i^{1+K}-P_i^{1-K}}{2K}, \qquad -1<K<1,$$8 are reconstructed numerically (Drepanou et al., 2021).

In the gravity–thermodynamics formulation, one works on the apparent horizon $$S_{K}=-\sum_{i=1}^{W}\frac{P_i^{1+K}-P_i^{1-K}}{2K}, \qquad -1<K<1,$$9 of a spatially flat FRW spacetime, assigns the temperature $K\to0$0, and imposes

$K\to0$1

Differentiating the Kaniadakis entropy and using the matter continuity equation yields

$K\to0$2

and, after integration,

$K\to0$3

with $K\to0$4. The extra terms are then identified with an effective dark-energy sector $K\to0$5 (Lymperis et al., 2021).

A third line of work uses the first-order Kaniadakis expansion together with the Hubble radius $K\to0$6. In one formulation this gives

$K\to0$7

with $K\to0$8, while in a modified Kaniadakis cosmology one writes

$K\to0$9

These models are frequently described as $P_i=1/W$0 dark-energy systems and are designed to capture the first-order Kaniadakis contribution when the Hubble horizon is the infrared cutoff (Fang et al., 2024, Sheykhi et al., 13 Oct 2025).

Extensions proliferate beyond these baseline cases. Early flat-FRW apparent-horizon KHDE wrote

$P_i=1/W$1

while non-flat models used either the apparent horizon $P_i=1/W$2 or a generalized future event horizon. In Brans–Dicke cosmology, the corresponding Hubble-cutoff density becomes

$P_i=1/W$3

so the effective gravitational coupling is carried by the scalar $P_i=1/W$4 rather than by a constant $P_i=1/W$5 (Moradpour et al., 2020, Sharma et al., 2021, Ghaffari, 2021).

3. Background dynamics and equation-of-state phenomenology

The future-event-horizon KHDE model reproduces the standard sequence of matter domination followed by dark-energy domination. In that construction, $P_i=1/W$6 at high redshift, the deceleration–acceleration transition occurs at $P_i=1/W$7, and the dark-energy equation of state can be quintessence-like, phantom-like, or exhibit phantom-divide crossing depending on the pair $P_i=1/W$8. In the far future, $P_i=1/W$9, and the asymptotic value of $W=\exp(S_{BH})$0 depends on the model parameters rather than being fixed universally to $W=\exp(S_{BH})$1 (Drepanou et al., 2021).

The horizon-first-law construction has a different phenomenology. There, the effective Kaniadakis dark-energy density and pressure are explicit functions of $W=\exp(S_{BH})$2 and $W=\exp(S_{BH})$3, the transition redshift is again around $W=\exp(S_{BH})$4, and for any $W=\exp(S_{BH})$5 the effective equation of state satisfies $W=\exp(S_{BH})$6 at low redshift while approaching $W=\exp(S_{BH})$7 from below in the asymptotic future. The same framework admits a $W=\exp(S_{BH})$8 limit, sometimes called pure holographic Kaniadakis dark energy, in which the K-deformation alone drives late-time acceleration; in that case $W=\exp(S_{BH})$9 implies $$S_{K}=\frac{1}{K}\sinh(KS_{BH}), \qquad S_{BH}=\frac{A}{4G}.$$0–$$S_{K}=\frac{1}{K}\sinh(KS_{BH}), \qquad S_{BH}=\frac{A}{4G}.$$1 (Lymperis et al., 2021).

Non-flat KHDE further broadens the classification of background histories. For the generalized future-event-horizon model, $$S_{K}=\frac{1}{K}\sinh(KS_{BH}), \qquad S_{BH}=\frac{A}{4G}.$$2 yields purely quintessence-like behavior, $$S_{K}=\frac{1}{K}\sinh(KS_{BH}), \qquad S_{BH}=\frac{A}{4G}.$$3 reproduces $$S_{K}=\frac{1}{K}\sinh(KS_{BH}), \qquad S_{BH}=\frac{A}{4G}.$$4CDM, and $$S_{K}=\frac{1}{K}\sinh(KS_{BH}), \qquad S_{BH}=\frac{A}{4G}.$$5 yields a quintom regime with crossing of $$S_{K}=\frac{1}{K}\sinh(KS_{BH}), \qquad S_{BH}=\frac{A}{4G}.$$6. Positive or negative curvature shifts the crossing threshold so that quintom behavior can occur for slightly smaller $$S_{K}=\frac{1}{K}\sinh(KS_{BH}), \qquad S_{BH}=\frac{A}{4G}.$$7 than in the flat case, while the deceleration–acceleration transition is only mildly shifted relative to the flat prediction $$S_{K}=\frac{1}{K}\sinh(KS_{BH}), \qquad S_{BH}=\frac{A}{4G}.$$8 (P et al., 2022).

The Hubble-cutoff branch changes a standard HDE expectation. In the modified Kaniadakis cosmology revisited later, the non-interacting model with $$S_{K}=\frac{1}{K}\sinh(KS_{BH}), \qquad S_{BH}=\frac{A}{4G}.$$9 can already explain the current acceleration, whereas in the interacting case the total equation of state $S_{K}=S_{BH}+\frac{K^2}{6}S_{BH}^3+\mathcal O(K^4)$0 can cross the phantom line at the present time. In the dark-energy-dominated limit, that model yields $S_{K}=S_{BH}+\frac{K^2}{6}S_{BH}^3+\mathcal O(K^4)$1 exactly and mimics a cosmological constant (Sheykhi et al., 13 Oct 2025). In the first-order $S_{K}=S_{BH}+\frac{K^2}{6}S_{BH}^3+\mathcal O(K^4)$2 realization, the matter-to-dark-energy equality occurs at $S_{K}=S_{BH}+\frac{K^2}{6}S_{BH}^3+\mathcal O(K^4)$3, $S_{K}=S_{BH}+\frac{K^2}{6}S_{BH}^3+\mathcal O(K^4)$4 and $S_{K}=S_{BH}+\frac{K^2}{6}S_{BH}^3+\mathcal O(K^4)$5 as $S_{K}=S_{BH}+\frac{K^2}{6}S_{BH}^3+\mathcal O(K^4)$6, so the model avoids a future big-rip singularity (Fang et al., 2024).

Taken together, these results show that there is no single KHDE equation-of-state signature. This suggests that statements such as “KHDE is necessarily phantom” or “KHDE is equivalent to standard HDE with a small correction” are only valid for specific implementations.

4. Observational constraints and statistical status

Observational analyses of the event-horizon KHDE model generally find that the Kaniadakis deformation is small. A joint MCMC analysis using cosmic chronometers, Pantheon supernovae, and transverse BAO gave

$S_{K}=S_{BH}+\frac{K^2}{6}S_{BH}^3+\mathcal O(K^4)$7

so that $S_{K}=S_{BH}+\frac{K^2}{6}S_{BH}^3+\mathcal O(K^4)$8 within errors. The same analysis reconstructed a deceleration–acceleration transition redshift

$S_{K}=S_{BH}+\frac{K^2}{6}S_{BH}^3+\mathcal O(K^4)$9

and found a matter-dominated past attractor and a dark-energy-dominated future attractor (Hernández-Almada et al., 2021).

For the Kaniadakis horizon-entropy cosmology derived from the first law, a broader joint dataset—OHD, Pantheon SNIa, HII galaxies, strong-lensing systems, and BAO—gave, in the $S_{BH}$0 case,

$S_{BH}$1

again consistent with $S_{BH}$2, while the $S_{BH}$3 case gave

$S_{BH}$4

The reconstructed transition redshift was

$S_{BH}$5

the $S_{BH}$6 diagnostic indicated a relaxation of the $S_{BH}$7 tension at low $S_{BH}$8, AICc/BIC/DIC found the model statistically equivalent to $S_{BH}$9CDM for many data combinations, and the BBN deviations were $S_K$0 for $S_K$1 or $S_K$2 for $S_K$3 (Hernández-Almada et al., 2021).

The first-order Hubble-cutoff realization yields a very different numerical fit. Using Pantheon SNIa+$S_K$4, one study obtained

$S_K$5

$S_K$6

and, through $S_K$7,

$S_K$8

That analysis reported an age $S_K$9 and interpreted the larger $\rho_{DE}L^4\lesssim S$0 as resolving the Hubble tension (Fang et al., 2024).

More recent late-time fits with CC, PantheonPlus or Union3, and DESI DR2 BAO again push the deformation toward the standard limit. Representative results include $\rho_{DE}L^4\lesssim S$1, $\rho_{DE}L^4\lesssim S$2, $\rho_{DE}L^4\lesssim S$3, and $\rho_{DE}L^4\lesssim S$4 for PP+BAO+CC, and $\rho_{DE}L^4\lesssim S$5, $\rho_{DE}L^4\lesssim S$6, $\rho_{DE}L^4\lesssim S$7 for U3+BAO+CC, with $\rho_{DE}L^4\lesssim S$8 indicating weak evidence favoring $\rho_{DE}L^4\lesssim S$9CDM or statistical equivalence depending on the dataset combination (Luciano et al., 22 Sep 2025). In a broader Bayesian-evidence comparison of generalized nonextensive HDE models, the Kaniadakis case was found to be strongly disfavored relative to flat $$S_{K}=-\sum_{i=1}^{W}\frac{P_i^{1+K}-P_i^{1-K}}{2K}, \qquad -1<K<1,$$00CDM, with

$$S_{K}=-\sum_{i=1}^{W}\frac{P_i^{1+K}-P_i^{1-K}}{2K}, \qquad -1<K<1,$$01

(Cimdiker et al., 23 Mar 2025).

The observational picture is therefore model-dependent rather than uniform. Event-horizon and horizon-first-law realizations often constrain the deformation close to zero, whereas Hubble-cutoff first-order models can prefer substantially different late-time backgrounds.

5. Dynamical systems, perturbative stability, and early-Universe consistency

Global dynamical analyses of Kaniadakis horizon-entropy cosmology use compact variables such as

$$S_{K}=-\sum_{i=1}^{W}\frac{P_i^{1+K}-P_i^{1-K}}{2K}, \qquad -1<K<1,$$02

or analogous compactifications. In this formulation, matter-dominated points $$S_{K}=-\sum_{i=1}^{W}\frac{P_i^{1+K}-P_i^{1-K}}{2K}, \qquad -1<K<1,$$03 are saddles or past repellers, de Sitter points are future attractors, and heteroclinic sequences connect matter domination to late-time de Sitter. The same analysis shows that the model does not admit bounce or turnaround solutions, because the required reversal-symmetry condition is not satisfied and direct numerical inspection shows that $$S_{K}=-\sum_{i=1}^{W}\frac{P_i^{1+K}-P_i^{1-K}}{2K}, \qquad -1<K<1,$$04 does not cross zero (Hernández-Almada et al., 2021).

The event-horizon KHDE dynamical system leads to the same qualitative cosmic ordering. With compact variables $$S_{K}=-\sum_{i=1}^{W}\frac{P_i^{1+K}-P_i^{1-K}}{2K}, \qquad -1<K<1,$$05 and $$S_{K}=-\sum_{i=1}^{W}\frac{P_i^{1+K}-P_i^{1-K}}{2K}, \qquad -1<K<1,$$06, the past attractor is the matter point $$S_{K}=-\sum_{i=1}^{W}\frac{P_i^{1+K}-P_i^{1-K}}{2K}, \qquad -1<K<1,$$07, the future attractor is a de Sitter sink $$S_{K}=-\sum_{i=1}^{W}\frac{P_i^{1+K}-P_i^{1-K}}{2K}, \qquad -1<K<1,$$08, and additional scaling points appear only for $$S_{K}=-\sum_{i=1}^{W}\frac{P_i^{1+K}-P_i^{1-K}}{2K}, \qquad -1<K<1,$$09. This reinforces the conclusion that Kaniadakis deformations preserve the standard matter-to-acceleration sequence at the background level (Hernández-Almada et al., 2021).

Classical stability is more model-sensitive. In non-flat KHDE, the squared sound speed $$S_{K}=-\sum_{i=1}^{W}\frac{P_i^{1+K}-P_i^{1-K}}{2K}, \qquad -1<K<1,$$10 is positive at late times for $$S_{K}=-\sum_{i=1}^{W}\frac{P_i^{1+K}-P_i^{1-K}}{2K}, \qquad -1<K<1,$$11 but the model develops instabilities in the deep past, while quintom solutions with $$S_{K}=-\sum_{i=1}^{W}\frac{P_i^{1+K}-P_i^{1-K}}{2K}, \qquad -1<K<1,$$12 are typically classically unstable during the entire evolution (P et al., 2022). In Brans–Dicke KHDE with Hubble cutoff, the non-interacting model has $$S_{K}=-\sum_{i=1}^{W}\frac{P_i^{1+K}-P_i^{1-K}}{2K}, \qquad -1<K<1,$$13 for all $$S_{K}=-\sum_{i=1}^{W}\frac{P_i^{1+K}-P_i^{1-K}}{2K}, \qquad -1<K<1,$$14, whereas an interaction $$S_{K}=-\sum_{i=1}^{W}\frac{P_i^{1+K}-P_i^{1-K}}{2K}, \qquad -1<K<1,$$15 with $$S_{K}=-\sum_{i=1}^{W}\frac{P_i^{1+K}-P_i^{1-K}}{2K}, \qquad -1<K<1,$$16 can produce $$S_{K}=-\sum_{i=1}^{W}\frac{P_i^{1+K}-P_i^{1-K}}{2K}, \qquad -1<K<1,$$17 over a finite period and render the model classically stable for some parameter ranges (Ghaffari, 2021). By contrast, in the later interacting KHDE model built inside modified Kaniadakis cosmology, $$S_{K}=-\sum_{i=1}^{W}\frac{P_i^{1+K}-P_i^{1-K}}{2K}, \qquad -1<K<1,$$18 throughout the interacting case, while the non-interacting case can be positive in the past but typically becomes negative at late times (Sheykhi et al., 13 Oct 2025).

These stability results imply that background viability does not automatically extend to perturbative viability. This is one of the central technical caveats in the KHDE literature.

6. Thermodynamic reinterpretations and later extensions

Thermodynamic analyses of the $$S_{K}=-\sum_{i=1}^{W}\frac{P_i^{1+K}-P_i^{1-K}}{2K}, \qquad -1<K<1,$$19 realization have emphasized that the trapping-horizon temperature is dynamically corrected. With $$S_{K}=-\sum_{i=1}^{W}\frac{P_i^{1+K}-P_i^{1-K}}{2K}, \qquad -1<K<1,$$20 and

$$S_{K}=-\sum_{i=1}^{W}\frac{P_i^{1+K}-P_i^{1-K}}{2K}, \qquad -1<K<1,$$21

the horizon temperature is written as

$$S_{K}=-\sum_{i=1}^{W}\frac{P_i^{1+K}-P_i^{1-K}}{2K}, \qquad -1<K<1,$$22

For the best-fit parameters in that model, $$S_{K}=-\sum_{i=1}^{W}\frac{P_i^{1+K}-P_i^{1-K}}{2K}, \qquad -1<K<1,$$23 remains finite, changes sign because the trapping horizon is an inner horizon, and tends to a constant in the deep future. The same analysis finds that the entropy-area law becomes

$$S_{K}=-\sum_{i=1}^{W}\frac{P_i^{1+K}-P_i^{1-K}}{2K}, \qquad -1<K<1,$$24

so the usual area law is rescaled and acquires an $$S_{K}=-\sum_{i=1}^{W}\frac{P_i^{1+K}-P_i^{1-K}}{2K}, \qquad -1<K<1,$$25 correction (Chen et al., 2024).

A more geometric thermodynamic treatment uses the Hayward–Kodama formalism. There the KHDE fluid with

$$S_{K}=-\sum_{i=1}^{W}\frac{P_i^{1+K}-P_i^{1-K}}{2K}, \qquad -1<K<1,$$26

induces a geometric equation of state

$$S_{K}=-\sum_{i=1}^{W}\frac{P_i^{1+K}-P_i^{1-K}}{2K}, \qquad -1<K<1,$$27

with $$S_{K}=-\sum_{i=1}^{W}\frac{P_i^{1+K}-P_i^{1-K}}{2K}, \qquad -1<K<1,$$28. The resulting horizon thermodynamics exhibits a Van der Waals type structure, an inverted first-order phase transition, and a non-physical swallowtail behavior in the Gibbs free energy. The same work reports that background data constrain only parameter combinations because of exact degeneracies, so perturbation-level information is required for independent bounds (Gonzalez-Espinoza et al., 22 Mar 2026).

KHDE has also been transplanted into broader modified-gravity settings. In 4D Einstein–Gauss–Bonnet gravity coupled to KHDE and fitted to CC and Pantheon+ data, the mean values

$$S_{K}=-\sum_{i=1}^{W}\frac{P_i^{1+K}-P_i^{1-K}}{2K}, \qquad -1<K<1,$$29

imply a phantom-like equation of state today, while the Gauss–Bonnet coupling remains consistent with $$S_{K}=-\sum_{i=1}^{W}\frac{P_i^{1+K}-P_i^{1-K}}{2K}, \qquad -1<K<1,$$30 within $$S_{K}=-\sum_{i=1}^{W}\frac{P_i^{1+K}-P_i^{1-K}}{2K}, \qquad -1<K<1,$$31 (Li et al., 26 Nov 2025). In Brans–Dicke complex quintessence, KHDE is likewise embedded in an interacting scalar-field sector, where $$S_{K}=-\sum_{i=1}^{W}\frac{P_i^{1+K}-P_i^{1-K}}{2K}, \qquad -1<K<1,$$32–$$S_{K}=-\sum_{i=1}^{W}\frac{P_i^{1+K}-P_i^{1-K}}{2K}, \qquad -1<K<1,$$33 and interactions can push $$S_{K}=-\sum_{i=1}^{W}\frac{P_i^{1+K}-P_i^{1-K}}{2K}, \qquad -1<K<1,$$34 closer to the $$S_{K}=-\sum_{i=1}^{W}\frac{P_i^{1+K}-P_i^{1-K}}{2K}, \qquad -1<K<1,$$35CDM point $$S_{K}=-\sum_{i=1}^{W}\frac{P_i^{1+K}-P_i^{1-K}}{2K}, \qquad -1<K<1,$$36 in the $$S_{K}=-\sum_{i=1}^{W}\frac{P_i^{1+K}-P_i^{1-K}}{2K}, \qquad -1<K<1,$$37 plane (Sadeghi et al., 2022).

Across these extensions, KHDE functions less as a single dark-energy ansatz than as an entropic template. Its defining ingredient is the replacement $$S_{K}=-\sum_{i=1}^{W}\frac{P_i^{1+K}-P_i^{1-K}}{2K}, \qquad -1<K<1,$$38; the observable consequences depend strongly on which horizon is used, how the cutoff is imposed, and whether the entropy deformation is interpreted as a modified fluid, a modified Friedmann equation, or a modification of the underlying gravitational sector.