Generalized MHR Entropy Cosmology

• The paper introduces a generalized mass-to-horizon scaling replacing the linear relation with Lⁿ, yielding modified entropy and Friedmann equations.
• It derives cosmological dynamics from the first-law of thermodynamics, integrating effective dark energy and a varying gravitational coupling.
• Observational and perturbative studies indicate that deviations from the standard Bekenstein-Hawking entropy are minimal, preserving ΛCDM-like behavior.

Generalized mass-to-horizon entropy-inspired modified cosmology denotes a class of horizon-thermodynamic cosmologies in which the apparent horizon of a Friedmann-Lemaître-Robertson-Walker universe is assigned a generalized entropy, and the cosmological field equations are obtained from a first-law or Clausius relation rather than postulated from an action alone. In its characteristic formulation, the standard linear horizon-mass scaling is replaced by a generalized mass-to-horizon relation $M=\gamma \frac{c^2}{G}L^n$, which induces a horizon entropy $$S_h=\gamma \frac{2n}{n+1}L^{\,n-1}S_{BH}\propto L^{n+1}$$; in parallel modified-gravity formulations, the same thermodynamic logic is expressed through an effective coupling $G_{\rm eff}$ entering both the Wald-Kodama entropy and generalized quasilocal masses. The resulting models modify the Friedmann equations, admit effective dark-energy or varying-$G$ interpretations, and are tested by background expansion, structure growth, primordial observables, and generalized second-law criteria (Gohar et al., 2023, Tian et al., 2014).

1. Thermodynamic basis at the apparent horizon

The common geometric setting is the FLRW spacetime, with apparent-horizon radius

$\tilde r_A=\frac{1}{\sqrt{H^2+k/a^2}}$

or, in flat space, $\tilde r_A=1/H$. The horizon area and volume are

$$A=4\pi \tilde r_A^2,\qquad V=\frac{4\pi}{3}\tilde r_A^3.$$

The temperature is typically taken to be either the positive Cai-Kim form $T=\frac{1}{2\pi \tilde r_A}$ or the Hayward-Kodama form

$$T_h=-\frac{1}{2\pi \tilde r_A}\left(1-\frac{\dot{\tilde r}_A}{2H\tilde r_A}\right),$$

depending on whether the treatment emphasizes quasi-static horizon thermodynamics or fully dynamical apparent-horizon thermodynamics (Tian et al., 2014, Prasanthan et al., 30 Jun 2026).

Within this framework, several equivalent-looking first-law relations are used. In standard entropic-cosmology derivations one writes $-dE=T\,dS$ for the heat flow through the horizon. In unified-horizon formulations one instead uses

$$S_h=\gamma \frac{2n}{n+1}L^{\,n-1}S_{BH}\propto L^{n+1}$$0

with work density $$S_h=\gamma \frac{2n}{n+1}L^{\,n-1}S_{BH}\propto L^{n+1}$$1. Tian and Booth formulated an even more general nonequilibrium version,

$$S_h=\gamma \frac{2n}{n+1}L^{\,n-1}S_{BH}\propto L^{n+1}$$2

where $$S_h=\gamma \frac{2n}{n+1}L^{\,n-1}S_{BH}\propto L^{n+1}$$3 is an energy-dissipation term induced by a time-dependent $$S_h=\gamma \frac{2n}{n+1}L^{\,n-1}S_{BH}\propto L^{n+1}$$4 in modified gravity, and argued that the horizon first law is the smooth limit of the interior unified first law as the integration sphere approaches the apparent horizon (Tian et al., 2014).

The material content is usually a perfect fluid obeying

$$S_h=\gamma \frac{2n}{n+1}L^{\,n-1}S_{BH}\propto L^{n+1}$$5

unless the generalized construction is rewritten as a varying-coupling theory or as an interacting effective sector. In those cases the continuity law is modified by the thermodynamic deformation itself. For example, in the $$S_h=\gamma \frac{2n}{n+1}L^{\,n-1}S_{BH}\propto L^{n+1}$$6-based formulation one has

$$S_h=\gamma \frac{2n}{n+1}L^{\,n-1}S_{BH}\propto L^{n+1}$$7

and the nonequilibrium correction is

$$S_h=\gamma \frac{2n}{n+1}L^{\,n-1}S_{BH}\propto L^{n+1}$$8

(Tian et al., 2014).

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

The defining move of the mass-to-horizon program is to generalize the horizon-associated mass from the standard linear scaling to

$$S_h=\gamma \frac{2n}{n+1}L^{\,n-1}S_{BH}\propto L^{n+1}$$9

with $G_{\rm eff}$0 and $G_{\rm eff}$1. Combining $G_{\rm eff}$2, the Hawking temperature $G_{\rm eff}$3, and the Clausius relation $G_{\rm eff}$4 yields

$G_{\rm eff}$5

Thus the entropy scales as $G_{\rm eff}$6, and the Bekenstein-Hawking law is recovered at $G_{\rm eff}$7 (Gohar et al., 2023).

This construction was introduced as a thermodynamically consistent alternative to simply postulating a generalized entropy while retaining both the Hawking temperature and a linear mass-to-horizon law. Later formulations generalized the relation further with additional deformation parameters $G_{\rm eff}$8, producing a master entropy family

$G_{\rm eff}$9

which was proposed as a thermodynamic seed for Bekenstein-Hawking, Tsallis-Cirto, Barrow, and quantum-gravity-corrected entropies (Gohar, 8 Oct 2025).

Several special identifications recur across the literature:

Parameter choice	Entropic identification	Cosmological note
$G$0	$G$1	standard limit
$G$2	standard Misner-Sharp mass in spherical symmetry	special normalization
$G$3	Tsallis-Cirto class	power-law deformation
$G$4	Barrow class	$G$5
$G$6	$G$7	exact $G$8CDM background in one formulation

The $G$9 case is especially notable. In the original generalized-MHR letter, mass scaling with horizon volume made the entropic density constant and the model exactly equivalent to $\tilde r_A=\frac{1}{\sqrt{H^2+k/a^2}}$0CDM (Gohar et al., 2023). Later horizon-first-law formulations, however, often write the modified Friedmann equation with explicit factors of $\tilde r_A=\frac{1}{\sqrt{H^2+k/a^2}}$1, so $\tilde r_A=\frac{1}{\sqrt{H^2+k/a^2}}$2 becomes a singular special case that must be handled separately rather than inserted directly into the integrated formula (Sheykhi, 19 Dec 2025).

3. Modified Friedmann equations and effective cosmological sectors

In the most widely used flat-FLRW implementation, the generalized entropy

$\tilde r_A=\frac{1}{\sqrt{H^2+k/a^2}}$3

leads to

$\tilde r_A=\frac{1}{\sqrt{H^2+k/a^2}}$4

and, after integration,

$\tilde r_A=\frac{1}{\sqrt{H^2+k/a^2}}$5

For $\tilde r_A=\frac{1}{\sqrt{H^2+k/a^2}}$6 these relations are commonly rewritten in Einstein-like form,

$\tilde r_A=\frac{1}{\sqrt{H^2+k/a^2}}$7

with

$\tilde r_A=\frac{1}{\sqrt{H^2+k/a^2}}$8

$\tilde r_A=\frac{1}{\sqrt{H^2+k/a^2}}$9

This makes the modification look like an emergent dark-energy sector rather than a direct alteration of the matter fluid (Basilakos et al., 31 Mar 2025).

A closely related formulation writes the modified background equation as

$\tilde r_A=1/H$0

Here the deformation is shifted to the geometric side: the Hubble-curvature combination is raised to a nonstandard power and the coupling is renormalized to $\tilde r_A=1/H$1 (Sheykhi, 19 Dec 2025).

The $\tilde r_A=1/H$2-based modified-gravity program predates the explicit generalized-MHR literature and provides a more general umbrella. There the horizon entropy is

$\tilde r_A=1/H$3

the generalized Misner-Sharp mass is

$\tilde r_A=1/H$4

and thermodynamic consistency is checked by demanding $\tilde r_A=1/H$5. In this setting the same effective coupling that multiplies the field equations also enters both the entropy and the quasilocal mass, producing a unified nonequilibrium thermodynamic derivation of the Friedmann equations across multiple modified-gravity theories (Tian et al., 2014).

4. Variants, reinterpretations, and theoretical structure

One important conceptual variant arises from Jacobson-style entropy-first gravity. If the horizon entropy is written as

$\tilde r_A=1/H$6

then the resulting field equations imply

$\tilde r_A=1/H$7

For Tsallis entropy this gives

$\tilde r_A=1/H$8

while a logarithmic correction produces

$\tilde r_A=1/H$9

In cosmology, taking $$A=4\pi \tilde r_A^2,\qquad V=\frac{4\pi}{3}\tilde r_A^3.$$0 to be the apparent-horizon area converts the entropy deformation into a horizon-scale-dependent effective Newton coupling and modifies the continuity law accordingly (Lu et al., 2024).

A second variant reformulates generalized entropy corrections as effective matter-sector modifications. In that approach the modified Friedmann equation is written as

$$A=4\pi \tilde r_A^2,\qquad V=\frac{4\pi}{3}\tilde r_A^3.$$1

with the generalized entropy encoded in a correction function $$A=4\pi \tilde r_A^2,\qquad V=\frac{4\pi}{3}\tilde r_A^3.$$2. The stress tensor remains perfect-fluid-like, but $$A=4\pi \tilde r_A^2,\qquad V=\frac{4\pi}{3}\tilde r_A^3.$$3 and $$A=4\pi \tilde r_A^2,\qquad V=\frac{4\pi}{3}\tilde r_A^3.$$4 are rescaled by the same factor $$A=4\pi \tilde r_A^2,\qquad V=\frac{4\pi}{3}\tilde r_A^3.$$5 (Khodam-Mohammadi et al., 2023).

A third line of work derives generalized entropy phenomenology from stochastic conformal metric fluctuations rather than postulating a new entropy from the outset. In that framework the averaged Friedmann equation becomes

$$A=4\pi \tilde r_A^2,\qquad V=\frac{4\pi}{3}\tilde r_A^3.$$6

and the correction is mapped to the generic entropy-deformed term

$$A=4\pi \tilde r_A^2,\qquad V=\frac{4\pi}{3}\tilde r_A^3.$$7

Rényi, dual Kaniadakis, Barrow, logarithmic, inverse-area, and MOND-inspired hypergeometric entropy corrections are then reproduced by different variance scalings $$A=4\pi \tilde r_A^2,\qquad V=\frac{4\pi}{3}\tilde r_A^3.$$8 (Khodahami et al., 18 Feb 2026).

These reinterpretations do not eliminate the structural tension within the subject. The generalized-MHR program was introduced precisely because several authors argued that generalized entropies combined with the Hawking temperature and an unmodified linear MHR collapse back to the Bekenstein-Hawking case. The later literature treats the generalized MHR as the minimal extra input needed to keep the Clausius relation intact while avoiding that degeneracy (Luciano et al., 18 Aug 2025).

5. Linear growth, nonlinear collapse, and early-Universe probes

The perturbative sector is a central discriminator because many entropy-modified backgrounds can mimic $$A=4\pi \tilde r_A^2,\qquad V=\frac{4\pi}{3}\tilde r_A^3.$$9CDM geometrically. In the Bekenstein MHEC case $T=\frac{1}{2\pi \tilde r_A}$0, the entropic component satisfies

$T=\frac{1}{2\pi \tilde r_A}$1

and the continuity equation implies interaction between the entropic sector and matter-radiation. When that interaction is perturbed consistently, the linear-growth equations reduce to a form in which $T=\frac{1}{2\pi \tilde r_A}$2 enters only through the background functions $T=\frac{1}{2\pi \tilde r_A}$3 and $T=\frac{1}{2\pi \tilde r_A}$4; numerical solutions then follow the $T=\frac{1}{2\pi \tilde r_A}$5CDM growth history within current growth uncertainties. If, instead, the interaction perturbation is neglected, extra damping terms appear and growth is artificially suppressed (Ali et al., 11 Jul 2025).

For general $T=\frac{1}{2\pi \tilde r_A}$6, a spherical top-hat analysis gives the linear matter-contrast equation

$T=\frac{1}{2\pi \tilde r_A}$7

with growing mode

$T=\frac{1}{2\pi \tilde r_A}$8

In that treatment, $T=\frac{1}{2\pi \tilde r_A}$9 enhances growth relative to $$T_h=-\frac{1}{2\pi \tilde r_A}\left(1-\frac{\dot{\tilde r}_A}{2H\tilde r_A}\right),$$0CDM, while $$T_h=-\frac{1}{2\pi \tilde r_A}\left(1-\frac{\dot{\tilde r}_A}{2H\tilde r_A}\right),$$1 suppresses it. The same modified background also affects primordial gravitational waves: $$T_h=-\frac{1}{2\pi \tilde r_A}\left(1-\frac{\dot{\tilde r}_A}{2H\tilde r_A}\right),$$2 suppresses the relic spectrum, whereas $$T_h=-\frac{1}{2\pi \tilde r_A}\left(1-\frac{\dot{\tilde r}_A}{2H\tilde r_A}\right),$$3 enhances it (Luciano, 1 Oct 2025).

A broader 2026 analysis extended the framework to cosmographic diagnostics, linear perturbations, halo mass functions, and cluster counts. It reported that the additional entropic correction changes the expansion history, the growth rate, and the abundance of collapsed halos, with more massive structures becoming less abundant and forming later. That work also stated that the $$T_h=-\frac{1}{2\pi \tilde r_A}\left(1-\frac{\dot{\tilde r}_A}{2H\tilde r_A}\right),$$4 branch passes a set of cosmographic null tests designed to discriminate it from flat and non-flat $$T_h=-\frac{1}{2\pi \tilde r_A}\left(1-\frac{\dot{\tilde r}_A}{2H\tilde r_A}\right),$$5CDM (Mondal et al., 16 Apr 2026).

Early-Universe probes generate a different class of constraints. In generalized-MHR baryogenesis, the entropy

$$T_h=-\frac{1}{2\pi \tilde r_A}\left(1-\frac{\dot{\tilde r}_A}{2H\tilde r_A}\right),$$6

produces modified background laws

$$T_h=-\frac{1}{2\pi \tilde r_A}\left(1-\frac{\dot{\tilde r}_A}{2H\tilde r_A}\right),$$7

This makes $$T_h=-\frac{1}{2\pi \tilde r_A}\left(1-\frac{\dot{\tilde r}_A}{2H\tilde r_A}\right),$$8 even during radiation domination and allows the usual gravitational-baryogenesis operator to generate a nonzero baryon asymmetry. For $$T_h=-\frac{1}{2\pi \tilde r_A}\left(1-\frac{\dot{\tilde r}_A}{2H\tilde r_A}\right),$$9, the paper finds

$-dE=T\,dS$0

equivalently $-dE=T\,dS$1 (Luciano et al., 3 Nov 2025).

6. Observational status, thermodynamic consistency, and open issues

Late-time background constraints currently do not support a single consensus picture. A DESI DR2 BAO analysis of the two-parameter GMHE model found, for the combined $-dE=T\,dS$2 dataset,

$-dE=T\,dS$3

with the exact $-dE=T\,dS$4CDM point $-dE=T\,dS$5 within about $-dE=T\,dS$6. Although the raw minimum $-dE=T\,dS$7 was comparable to or slightly better than $-dE=T\,dS$8CDM, the Akaike Information Criterion mildly favored $-dE=T\,dS$9CDM in every dataset combination (Luciano et al., 18 Aug 2025).

A later joint analysis including Pantheon+ with SH0ES calibration, DESI DR2 BAO, CMB distance priors, and growth data reported the opposite model-selection trend. For fixed-$$S_h=\gamma \frac{2n}{n+1}L^{\,n-1}S_{BH}\propto L^{n+1}$$00 families it found $$S_h=\gamma \frac{2n}{n+1}L^{\,n-1}S_{BH}\propto L^{n+1}$$01 values between $$S_h=\gamma \frac{2n}{n+1}L^{\,n-1}S_{BH}\propto L^{n+1}$$02 and $$S_h=\gamma \frac{2n}{n+1}L^{\,n-1}S_{BH}\propto L^{n+1}$$03 relative to $$S_h=\gamma \frac{2n}{n+1}L^{\,n-1}S_{BH}\propto L^{n+1}$$04CDM, while fixed-coupling scans showed strong preference for weak coupling, roughly

$$S_h=\gamma \frac{2n}{n+1}L^{\,n-1}S_{BH}\propto L^{n+1}$$05

In that regime the model becomes nearly degenerate with $$S_h=\gamma \frac{2n}{n+1}L^{\,n-1}S_{BH}\propto L^{n+1}$$06CDM in background parameters, but the combined evidence was reported as slightly to moderately favoring the entropic construction (Denkiewicz et al., 26 Dec 2025).

Background-only viability analyses are stricter than many early entropic models suggested. In the Cai-Kim formulation built from the generalized MHR with additional $$S_h=\gamma \frac{2n}{n+1}L^{\,n-1}S_{BH}\propto L^{n+1}$$07 deformations, phenomenologically acceptable scenarios were found to occupy only a narrow neighborhood of the Bekenstein-Hawking law. Power-law entanglement corrections can generate moderate early dark energy only in a tightly restricted region of parameter space, quantum-gravity corrections are Planck-suppressed and observationally irrelevant, and all viable models predict a $$S_h=\gamma \frac{2n}{n+1}L^{\,n-1}S_{BH}\propto L^{n+1}$$08CDM-like present-day background (Prasanthan et al., 30 Jun 2026).

The thermodynamic consistency program is more favorable than the observational one. In generalized mass-to-horizon entropy and horizon-thermodynamics studies, the entropy evolution satisfies

$$S_h=\gamma \frac{2n}{n+1}L^{\,n-1}S_{BH}\propto L^{n+1}$$09

at late times, indicating approach to a maximum-entropy state, and the fluctuation analysis gives finite horizon-energy variance

$$S_h=\gamma \frac{2n}{n+1}L^{\,n-1}S_{BH}\propto L^{n+1}$$10

with relative fluctuations decaying as the horizon degrees of freedom grow (Shameeem et al., 12 May 2026). In the emergent-space formulation, the generalized second law was also shown to hold for a universe bounded by the apparent horizon (Sheykhi, 19 Dec 2025).

Several open issues remain structural rather than numerical. The $$S_h=\gamma \frac{2n}{n+1}L^{\,n-1}S_{BH}\propto L^{n+1}$$11 branch is exactly $$S_h=\gamma \frac{2n}{n+1}L^{\,n-1}S_{BH}\propto L^{n+1}$$12CDM in one formulation but singular in later integrated equations. Entropy-first Jacobson constructions imply horizon-area-dependent field equations and quasi-local $$S_h=\gamma \frac{2n}{n+1}L^{\,n-1}S_{BH}\propto L^{n+1}$$13, raising questions about horizon choice and covariance. More generally, the field remains split between formulations that rewrite the entropy correction as effective dark energy, as a varying gravitational coupling, or as a deeper change in the relation between quasilocal mass and horizon information. What is common to all branches is narrower than often assumed: once thermodynamic consistency, early-universe bounds, background evolution, and perturbations are imposed simultaneously, phenomenologically acceptable departures from the Bekenstein-Hawking area law are typically small (Lu et al., 2024, Prasanthan et al., 30 Jun 2026).