Generalized Mass-to-Horizon Entropy

• The paper introduces a generalized mass-to-horizon relation that derives entropy from a nonlinear power-law mass-horizon scale, unifying thermodynamics with gravitational dynamics.
• It explains how the horizon entropy, scaling as L^(n+1), modifies Friedmann cosmology to yield an effective dark-energy sector without invoking extra fluids.
• It emphasizes that observational constraints force deviations to be minimal, recovering standard Bekenstein–Hawking entropy when n and γ approach unity.

Searching arXiv for the most relevant GMHE papers and related thermodynamic-gravity context. Generalized mass-to-horizon entropy is a class of horizon-entropy constructions in which the entropy assigned to a cosmological or black-hole horizon is not postulated independently, but is derived from a generalized relation between the horizon-associated mass and the horizon scale, together with the Clausius relation and a Hawking-type temperature. In the simplest and most widely used formulation, the generalized mass-to-horizon relation is written as $M=\gamma \frac{c^2}{G}L^n$, with $L$ the horizon scale, $n$ a real exponent, and $\gamma$ a normalization parameter; the associated entropy then scales as $S\propto L^{n+1}$, reproducing the Bekenstein–Hawking area law in the appropriate limit (Gohar et al., 2023, Basilakos et al., 31 Mar 2025). Within cosmology, this construction has been developed as a thermodynamically consistent extension of horizon entropy that modifies Friedmann dynamics through the gravity-thermodynamics conjecture, often yielding an effective dark-energy sector without introducing a new fluid by hand (Luciano et al., 18 Aug 2025). Subsequent work has emphasized both its phenomenological viability near the standard limit and the strong restrictions imposed by background evolution, structure growth, baryogenesis, and thermodynamic consistency (Prasanthan et al., 30 Jun 2026, Ali et al., 11 Jul 2025, Luciano et al., 3 Nov 2025, Shameeem et al., 12 May 2026).

1. Conceptual definition and thermodynamic motivation

Generalized mass-to-horizon entropy is founded on the claim that, in horizon thermodynamics, the primary object to generalize is the mass-to-horizon relation rather than the entropy functional alone. In the standard construction, one combines the Bekenstein–Hawking entropy

$S_{BH}=\frac{A}{4}, \qquad A=4\pi \tilde r_A^2,$

or in conventional units $S_{BH}=k_B c^3 A/(4\hbar G)$, with a Hawking-like temperature and a linear horizon-mass scaling. In the generalized framework, that linear scaling is replaced by a nonlinear one, and the entropy is then derived from thermodynamic consistency rather than imposed ad hoc (Luciano et al., 18 Aug 2025, Gohar, 8 Oct 2025).

A recurring motivation in the literature is the observation that many generalized entropy proposals cease to generate genuinely new cosmological dynamics if one simultaneously insists on the Clausius relation and retains a linear mass-to-horizon relation. In that case, generalized entropic cosmologies can become effectively degenerate with the standard Bekenstein–Hawking setup, inheriting the same limitations for late-time acceleration (Luciano et al., 18 Aug 2025, Luciano, 1 Oct 2025). The generalized mass-to-horizon construction was introduced specifically to break that degeneracy.

In this program, thermodynamic consistency means that the triplet $(M,T_h,S_h)$ is chosen so that the Clausius identity holds by construction. Depending on the formulation, this is written as

$dE=c^2\,dM=T_h\,dS_h,$

or, in cosmological apparent-horizon thermodynamics,

$dU=T_h\,dS-\mathcal W\,dV,$

with $L$0, $L$1, and $L$2 the work density (Luciano et al., 18 Aug 2025, Prasanthan et al., 30 Jun 2026). This distinguishes generalized mass-to-horizon entropy from approaches that simply replace $L$3 by some $L$4 while leaving the other thermodynamic ingredients untouched.

2. Generalized mass-to-horizon relation and entropy formula

The most studied version of the generalized mass-to-horizon relation is

$L$5

where $L$6 is the effective mass associated with the system, $L$7 is the horizon length scale, $L$8 is a non-negative real parameter controlling the power-law scaling, and $L$9 is a constant with dimensions $n$0 (Luciano et al., 18 Aug 2025, Basilakos et al., 31 Mar 2025). In cosmological applications, the relevant horizon is usually the apparent horizon, $n$1, while some entropic-cosmology formulations use the Hubble horizon $n$2 in a flat universe (Denkiewicz et al., 26 Dec 2025, Sheykhi, 19 Dec 2025).

Combining this generalized mass law with the Hawking temperature

$n$3

or, in conventional units,

$n$4

and the Clausius relation yields the generalized entropy

$n$5

equivalently

$n$6

so that $n$7 (Gohar et al., 2023, Luciano et al., 18 Aug 2025). In black-hole or apparent-horizon language this is also written as

$n$8

(Mondal et al., 23 Jun 2026).

The standard Bekenstein–Hawking entropy is recovered at

$n$9

for which the generalized mass-to-horizon relation becomes linear and the entropy reduces exactly to $\gamma$0 (Luciano et al., 18 Aug 2025, Basilakos et al., 31 Mar 2025). Several works assume physically viable departures satisfy

$\gamma$1

(Luciano et al., 18 Aug 2025).

A more elaborate extension replaces the simple power law by

$\gamma$2

which induces, under the approximation

$\gamma$3

the generalized entropy

$\gamma$4

(Prasanthan et al., 30 Jun 2026, Gohar, 8 Oct 2025). This extended form is presented as a unifying framework for corrected Tsallis–Cirto, Barrow, entanglement, and logarithmic quantum-gravity entropies (Gohar, 8 Oct 2025).

3. Cosmological horizon thermodynamics and modified Friedmann dynamics

The principal cosmological arena for generalized mass-to-horizon entropy is an FLRW universe bounded by the apparent horizon. In the spatially flat case,

$\gamma$5

and the cosmic fluid is modeled as a perfect fluid obeying

$\gamma$6

The work density is

$\gamma$7

and the thermodynamic identity at the apparent horizon is taken as

$\gamma$8

or, in related formulations,

$\gamma$9

(Luciano et al., 18 Aug 2025, Prasanthan et al., 30 Jun 2026).

With the Bekenstein–Hawking entropy, this reproduces the standard Friedmann equations. Replacing the entropy by the generalized mass-to-horizon entropy modifies the derivation. In one widely used formulation, the resulting cosmological dynamics can be recast as

$S\propto L^{n+1}$0

with effective dark-energy density and pressure

$S\propto L^{n+1}$1

$S\propto L^{n+1}$2

(Luciano et al., 18 Aug 2025, Basilakos et al., 31 Mar 2025). In this interpretation, the generalized entropy acts as a source of effective dark energy; no extra physical dark-energy fluid is introduced by hand.

The $S\propto L^{n+1}$3CDM limit is explicitly recovered for

$S\propto L^{n+1}$4

At that point, the generalized MHR becomes linear, the entropy becomes exactly Bekenstein–Hawking, and the modified Friedmann equations reduce to standard $S\propto L^{n+1}$5CDM (Luciano et al., 18 Aug 2025).

A separate line of work formulates the dynamics directly as a modified Friedmann equation of the form

$S\propto L^{n+1}$6

derived from the first law on the apparent horizon and reproduced independently from Padmanabhan’s emergent-space proposal (Sheykhi, 19 Dec 2025). This suggests that the same generalized entropy can be embedded both in equilibrium horizon thermodynamics and in emergent-gravity-inspired cosmic space dynamics.

4. Relations to other generalized entropies and modified-gravity interpretations

Generalized mass-to-horizon entropy is often presented as a unifying umbrella for several known entropy modifications. The mapping is typically expressed through the entropy’s power-law dependence on the horizon radius or area.

For the simple $S\propto L^{n+1}$7 model, the entropy scales as

$S\propto L^{n+1}$8

which reproduces several familiar cases:

• $S\propto L^{n+1}$9: Bekenstein entropy (Gohar et al., 2023).
• $S_{BH}=\frac{A}{4}, \qquad A=4\pi \tilde r_A^2,$0: Tsallis–Cirto entropy (Gohar et al., 2023, Mondal et al., 23 Jun 2026).
• $S_{BH}=\frac{A}{4}, \qquad A=4\pi \tilde r_A^2,$1: Barrow entropy (Gohar et al., 2023, Mondal et al., 23 Jun 2026).
• $S_{BH}=\frac{A}{4}, \qquad A=4\pi \tilde r_A^2,$2: Tsallis–Zamora entropy (Mondal et al., 23 Jun 2026).

The extended $S_{BH}=\frac{A}{4}, \qquad A=4\pi \tilde r_A^2,$3 formulation was explicitly designed to encompass the standard area-law extension with $S_{BH}=\frac{A}{4}, \qquad A=4\pi \tilde r_A^2,$4, power-law entanglement corrections with $S_{BH}=\frac{A}{4}, \qquad A=4\pi \tilde r_A^2,$5 and $S_{BH}=\frac{A}{4}, \qquad A=4\pi \tilde r_A^2,$6, and logarithmic quantum-gravity corrections in the $S_{BH}=\frac{A}{4}, \qquad A=4\pi \tilde r_A^2,$7 limit (Prasanthan et al., 30 Jun 2026, Gohar, 8 Oct 2025).

There are also two distinct modified-gravity reinterpretations of generalized horizon entropy in the literature. One line, based on Jacobson’s thermodynamic derivation of gravity, argues that a generalized entropy $S_{BH}=\frac{A}{4}, \qquad A=4\pi \tilde r_A^2,$8 implies an area-dependent effective gravitational coupling

$S_{BH}=\frac{A}{4}, \qquad A=4\pi \tilde r_A^2,$9

so generalized entropy should be understood as inducing modified gravitational dynamics rather than merely altering the black-hole first law inside unmodified GR (Lu et al., 2024, Wu et al., 22 Mar 2026). In this picture, the proper logic is generalized entropy $S_{BH}=k_B c^3 A/(4\hbar G)$0 modified gravity $S_{BH}=k_B c^3 A/(4\hbar G)$1 compatible horizon solution $S_{BH}=k_B c^3 A/(4\hbar G)$2 thermodynamics.

A second line reconstructs generalized mass-to-horizon entropy from the Iyer–Wald formalism in $S_{BH}=k_B c^3 A/(4\hbar G)$3 gravity. In that approach, the generalized entropy

$S_{BH}=k_B c^3 A/(4\hbar G)$4

can be reproduced by a power-law modified gravity Lagrangian of the form

$S_{BH}=k_B c^3 A/(4\hbar G)$5

so the entropic exponent $S_{BH}=k_B c^3 A/(4\hbar G)$6 is reinterpreted as a small deviation from general relativity (Mondal et al., 23 Jun 2026). This suggests that the generalized entropy may admit a covariant gravitational origin rather than being purely phenomenological.

5. Observational status and phenomenological constraints

The late-time observational status of generalized mass-to-horizon entropy is mixed but consistently points toward small deviations from the standard area law. A dedicated analysis using Type Ia supernovae, cosmic chronometers, BAO including DESI DR2, and the SH0ES prior on $S_{BH}=k_B c^3 A/(4\hbar G)$7 found that the extended entropic scenario gives a slightly better or statistically comparable fit to the data, but the Akaike Information Criterion mildly favors $S_{BH}=k_B c^3 A/(4\hbar G)$8CDM because the improvement in $S_{BH}=k_B c^3 A/(4\hbar G)$9 is too small to compensate for the two extra parameters (Luciano et al., 18 Aug 2025).

That study sampled

$(M,T_h,S_h)$0

and found, for the full $(M,T_h,S_h)$1 combination,

$(M,T_h,S_h)$2

with

$(M,T_h,S_h)$3

and the conclusion that the standard limit $(M,T_h,S_h)$4 lies within about $(M,T_h,S_h)$5, indicating no significant departure from standard cosmology with current data (Luciano et al., 18 Aug 2025).

An earlier background analysis with SNIa, CC, and BAO found

$(M,T_h,S_h)$6

for the $(M,T_h,S_h)$7 combination and concluded that the scenario is in agreement with observations, though the preferred value again lies close to the standard entropy law (Basilakos et al., 31 Mar 2025).

A broader observational analysis that included Pantheon+ with SH0ES, DESI DR2 BAO, compressed CMB distance priors, and structure-growth data reported Bayesian evidence slightly favoring weakly coupled mass-to-horizon entropic cosmologies over $(M,T_h,S_h)$8CDM when $(M,T_h,S_h)$9, while strong coupling such as $dE=c^2\,dM=T_h\,dS_h,$0 was decisively disfavored (Denkiewicz et al., 26 Dec 2025). This suggests that current data, when interpreted in that framework, prefer a weakly coupled entropic sector close to $dE=c^2\,dM=T_h\,dS_h,$1CDM.

A different, model-independent background analysis of the more general $dE=c^2\,dM=T_h\,dS_h,$2 formulation concluded even more sharply that cosmological viability restricts admissible deviations from the Bekenstein–Hawking law to a narrow neighborhood of the standard entropy. In the pure area-law extension with $dE=c^2\,dM=T_h\,dS_h,$3, viable models at fixed $dE=c^2\,dM=T_h\,dS_h,$4 were confined to

$dE=c^2\,dM=T_h\,dS_h,$5

while in the pure $dE=c^2\,dM=T_h\,dS_h,$6-rescaled Bekenstein case $dE=c^2\,dM=T_h\,dS_h,$7,

$dE=c^2\,dM=T_h\,dS_h,$8

(Prasanthan et al., 30 Jun 2026). That work emphasized that background cosmology alone imposes stringent restrictions on thermodynamically consistent generalized entropy constructions of this class.

6. Structure growth, early-universe probes, and thermodynamic viability

Beyond the background expansion, generalized mass-to-horizon entropy has been tested against several dynamical and thermodynamic criteria.

A perturbative study of generalized mass-to-horizon entropic cosmology focused on the Bekenstein case $dE=c^2\,dM=T_h\,dS_h,$9 and derived linear perturbation equations in the Newtonian sub-Hubble approximation (Ali et al., 11 Jul 2025). It distinguished between a fully perturbed interaction source and the common approximation in which the interaction perturbation is neglected. In the thermodynamically consistent implementation, where the interaction perturbation is included, the matter-growth history follows $dU=T_h\,dS-\mathcal W\,dV,$0CDM within current uncertainties, with the global minimum close to

$dU=T_h\,dS-\mathcal W\,dV,$1

and values as small as

$dU=T_h\,dS-\mathcal W\,dV,$2

being observationally indistinguishable from $dU=T_h\,dS-\mathcal W\,dV,$3CDM (Ali et al., 11 Jul 2025). That study also stressed that radiation perturbations are essential, since setting $dU=T_h\,dS-\mathcal W\,dV,$4 can underestimate matter growth by more than two orders of magnitude in that setup.

A later investigation of the generalized $dU=T_h\,dS-\mathcal W\,dV,$5 framework showed that linear matter perturbations in the spherical Top-Hat formalism are strongly influenced by the modified background dynamics: $dU=T_h\,dS-\mathcal W\,dV,$6 enhances growth relative to $dU=T_h\,dS-\mathcal W\,dV,$7CDM, whereas $dU=T_h\,dS-\mathcal W\,dV,$8 suppresses it (Luciano, 1 Oct 2025). The same work analyzed primordial gravitational waves and found that $dU=T_h\,dS-\mathcal W\,dV,$9 enhances the relic PGW spectrum while $L$00 suppresses it, leading to the approximate current lower bound

$L$01

from PTA exclusions in the nHz band (Luciano, 1 Oct 2025).

Baryogenesis provides an independent early-universe constraint. In a radiation-dominated background, standard GR gives $L$02, preventing the usual gravitational baryogenesis operator from generating an asymmetry. In the generalized mass-to-horizon entropy cosmology, the modified Friedmann equations make $L$03 even during radiation domination, so standard gravitational baryogenesis can operate (Luciano et al., 3 Nov 2025). Expanding near $L$04, the predicted asymmetry is

$L$05

with $L$06, yielding the bound

$L$07

for a benchmark decoupling scale $L$08 (Luciano et al., 3 Nov 2025). This points to a small sub-extensive deviation from the Bekenstein–Hawking area law.

Thermodynamic viability has also been studied directly. One analysis derived the entropy balance relation

$L$09

for generalized mass-to-horizon entropy, separating the thermal entropy flux from an internal entropy-production term, and found that the generalized second law holds for $L$10 while the entropy tends toward a stable maximum in the asymptotic de Sitter regime (Shameeem et al., 12 May 2026). The same work studied horizon-energy fluctuations and concluded that the relative fluctuation becomes extremely small at late times, supporting thermal stability (Shameeem et al., 12 May 2026). Another paper combining the first-law and emergent-space derivations also found that the generalized second law is fulfilled for the universe enveloped by the apparent horizon (Sheykhi, 19 Dec 2025).

7. Interpretive issues, caveats, and current outlook

Several caveats recur throughout the literature. First, the generalized mass-to-horizon relation itself is usually introduced as a phenomenological ansatz rather than derived from first principles. Although special cases such as the linear relation can be tied to the Misner–Sharp mass, the deeper geometric origin of the nonlinear scaling remains open in most cosmological formulations (Sheykhi, 19 Dec 2025, Shameeem et al., 12 May 2026).

Second, the framework is often developed specifically for FLRW cosmology with the apparent horizon and an equilibrium or quasi-equilibrium Clausius relation. Results obtained in this setting are not automatically general non-FLRW statements (Prasanthan et al., 30 Jun 2026). Third, several studies emphasize that current evidence for deviations from the standard area law remains weak: late-time data generally allow the $L$11CDM limit within $L$12, and background viability often collapses the admissible parameter space to a tiny neighborhood of the Bekenstein–Hawking case (Luciano et al., 18 Aug 2025, Prasanthan et al., 30 Jun 2026).

There are also conceptual differences between thermodynamic and modified-gravity readings of the framework. One interpretation treats generalized mass-to-horizon entropy as an effective thermodynamic description of cosmic acceleration, with dark energy emergent from horizon thermodynamics (Basilakos et al., 31 Mar 2025). Another interpretation holds that any generalized entropy must first imply modified gravitational dynamics, often encoded as an area-dependent or curvature-dependent effective coupling, before one discusses horizon thermodynamics (Lu et al., 2024, Mondal et al., 23 Jun 2026). These are not necessarily incompatible, but they represent different emphases.

A plausible implication is that generalized mass-to-horizon entropy is best viewed not as a single finished theory but as a thermodynamically constrained family of horizon-based modifications whose physically acceptable region is very close to the standard area law. This suggestion is consistent with the sharp background constraints found in the extended MHR framework and with the observational result that all viable models are effectively $L$13CDM-like at the present epoch (Prasanthan et al., 30 Jun 2026, Luciano et al., 18 Aug 2025).

At present, the most robust claims are the following. Generalized mass-to-horizon entropy provides a thermodynamically consistent route to generalized horizon entropies by promoting the mass-to-horizon relation to the primary object (Gohar et al., 2023, Gohar, 8 Oct 2025). It modifies Friedmann dynamics through the gravity-thermodynamics conjecture and can generate an effective dark-energy sector without postulating a new fluid (Luciano et al., 18 Aug 2025). It encompasses several known generalized entropies as limiting cases and may admit a modified-gravity foundation via Jacobson or Iyer–Wald constructions (Lu et al., 2024, Mondal et al., 23 Jun 2026). But all currently viable realizations lie close to the Bekenstein–Hawking limit, and neither late-time observations nor early-universe constraints presently require a statistically significant departure from standard cosmology (Luciano et al., 18 Aug 2025, Prasanthan et al., 30 Jun 2026).