---
title: Generalized MHR Entropy Cosmology
url: https://www.emergentmind.com/topics/generalized-mass-to-horizon-entropy-inspired-modified-cosmology
type: topic
---

# Generalized MHR Entropy Cosmology

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 [2307.06239][1409.4278].

## 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 [1409.4278][2607.00133].

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
\[
dE=T_h\,dS_h+W\,dV,
\]
with work density \(W=(\rho-p)/2\). Tian and Booth formulated an even more general nonequilibrium version,
\[
dE=A\psi+W\,dV+\mathcal E,
\]
where \(\mathcal E\) is an energy-dissipation term induced by a time-dependent \(G_{\rm eff}\) 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 [1409.4278].

The material content is usually a perfect fluid obeying
\[
\dot\rho+3H(\rho+p)=0,
\]
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 \(G_{\rm eff}\)-based formulation one has
\[
\dot\rho_{\rm eff}+3H(\rho_{\rm eff}+P_{\rm eff})
=-\frac{\dot G_{\rm eff}}{G_{\rm eff}}\rho_{\rm eff},
\]
and the nonequilibrium correction is
\[
\mathcal E=-\frac{4\pi}{3}\Upsilon^3\frac{\dot G_{\rm eff}}{G_{\rm eff}}\rho_{\rm eff}\,dt
\]
[1409.4278].

## 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
\[
M=\gamma \frac{c^2}{G}L^n,
\]
with \(n\ge 0\) and \([\gamma]=[L]^{1-n}\). Combining \(E=Mc^2\), the Hawking temperature \(T_{BH}=\frac{\hbar c}{2\pi k_B L}\), and the Clausius relation \(dE=T\,dS\) yields
\[
S_n=\gamma \frac{n}{n+1}\frac{2\pi k_B c^3}{G\hbar}L^{n+1}
=\gamma \frac{2n}{n+1}L^{n-1}S_{BH}.
\]
Thus the entropy scales as \(S_n\propto L^{n+1}\), and the Bekenstein-Hawking law is recovered at \(n=1,\gamma=1\) [2307.06239].

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 \((m,\beta,\alpha)\), producing a master entropy family
\[
S_G=
2\pi k_B\gamma\left[
\frac{m}{m+1}\left(\frac{L}{\ell_{\rm Pl}}\right)^{m+1}
\mp
m\frac{\sigma-1}{\sigma}\beta\left(\frac{L}{\ell_{\rm Pl}}\right)^\sigma
\right],
\qquad \sigma=m+3-\alpha,
\]
which was proposed as a thermodynamic seed for Bekenstein-Hawking, Tsallis-Cirto, Barrow, and quantum-gravity-corrected entropies [2510.07587].

Several special identifications recur across the literature:

| Parameter choice | Entropic identification | Cosmological note |
|---|---|---|
| \(n=1,\ \gamma=1\) | \(S_h=S_{BH}\) | standard limit |
| \(n=1,\ \gamma=\tfrac12\) | standard Misner-Sharp mass in spherical symmetry | special normalization |
| \(n=2\delta-1\) | Tsallis-Cirto class | power-law deformation |
| \(n=1+\Delta\) | Barrow class | \(1\le n\le 2\) |
| \(n=3\) | \(M\propto L^3,\ S\propto L^4\) | exact \(\Lambda\)CDM background in one formulation |

The \(n=3\) 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 \(\Lambda\)CDM [2307.06239]. Later horizon-first-law formulations, however, often write the modified Friedmann equation with explicit factors of \((3-n)^{-1}\), so \(n=3\) becomes a singular special case that must be handled separately rather than inserted directly into the integrated formula [2512.18054].

## 3. Modified Friedmann equations and effective cosmological sectors

In the most widely used flat-FLRW implementation, the generalized entropy
\[
S_n=\gamma \frac{2n}{n+1}r_a^{\,n-1}S_{BH}
\]
leads to
\[
-4\pi G(\rho_m+p_m)=
\gamma n\left(H^2+\frac{k}{a^2}\right)^{\frac{1-n}{2}}
\left(\dot H-\frac{k}{a^2}\right),
\]
and, after integration,
\[
\frac{8\pi G}{3}\rho_m=
\frac{2\gamma n}{3-n}\left(H^2+\frac{k}{a^2}\right)^{\frac{3-n}{2}}
-\frac{\Lambda}{3}.
\]
For \(k=0\) these relations are commonly rewritten in Einstein-like form,
\[
H^2=\frac{8\pi G}{3}\left(\rho_m+\rho_{DE}\right),\qquad
\dot H=-4\pi G\left(\rho_m+p_m+\rho_{DE}+p_{DE}\right),
\]
with
\[
\rho_{DE}=\frac{3}{8\pi G}\left[\frac{\Lambda}{3}+H^2-\frac{2\gamma n}{3-n}H^{3-n}\right],
\]
\[
p_{DE}=-\frac{1}{8\pi G}\left[\Lambda+(2\dot H+3H^2)-2\gamma n H^{1-n}\left(\dot H+\frac{3}{3-n}H^2\right)\right].
\]
This makes the modification look like an emergent dark-energy sector rather than a direct alteration of the matter fluid [2503.24355].

A closely related formulation writes the modified background equation as
\[
\left(H^2+\frac{k}{a^2}\right)^{(3-n)/2}
=\frac{8\pi G_{\rm eff}}{3}(\rho+\rho_\Lambda),
\qquad
G_{\rm eff}=\frac{(3-n)G}{2n\gamma}.
\]
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 \(G_{\rm eff}\) [2512.18054].

The \(G_{\rm eff}\)-based modified-gravity program predates the explicit generalized-MHR literature and provides a more general umbrella. There the horizon entropy is
\[
S=\frac{A_A}{4G_{\rm eff}},
\]
the generalized Misner-Sharp mass is
\[
M=\frac{\Upsilon^3}{2G_{\rm eff}}\left(H^2+\frac{k}{a^2}\right),
\]
and thermodynamic consistency is checked by demanding \(M=\rho_{\rm eff}V\). 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 [1409.4278].

## 4. Variants, reinterpretations, and theoretical structure

One important conceptual variant arises from Jacobson-style entropy-first gravity. If the horizon entropy is written as
\[
S=\frac{F(A)}{4G},
\]
then the resulting field equations imply
\[
G_{\rm eff}=\frac{G}{F'(A)}.
\]
For Tsallis entropy this gives
\[
G_{\rm eff}(A)=\frac{G}{\delta}\left(\frac{A}{A_0}\right)^{1-\delta},
\]
while a logarithmic correction produces
\[
G_{\rm eff}=\frac{G}{1+\dfrac{4C_1G}{A}}.
\]
In cosmology, taking \(A\) to be the apparent-horizon area converts the entropy deformation into a horizon-scale-dependent effective Newton coupling and modifies the continuity law accordingly [2407.00484].

A second variant reformulates generalized entropy corrections as effective matter-sector modifications. In that approach the modified Friedmann equation is written as
\[
H^2+\frac{k}{a^2}=\frac{8\pi}{3}\rho f(\rho),\qquad
\dot H-\frac{k}{a^2}=-4\pi(\rho+p)f(\rho),
\]
with the generalized entropy encoded in a correction function \(f(\rho)\). The stress tensor remains perfect-fluid-like, but \(\rho\) and \(p\) are rescaled by the same factor \(f(\rho)\) [2305.13339].

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
\[
\frac{8\pi G}{3}\langle \rho\rangle
=
H^2+\epsilon^2\left(H^2\langle b^2\rangle+\frac{\langle \dot b^2\rangle}{4}\right),
\]
and the correction is mapped to the generic entropy-deformed term
\[
-\int \frac{2\,dR}{R^3}\,\frac{d\delta S}{dS_{BH}}.
\]
Rényi, dual Kaniadakis, Barrow, logarithmic, inverse-area, and MOND-inspired hypergeometric entropy corrections are then reproduced by different variance scalings \(\langle b^2\rangle(S_{BH})\) [2602.16294].

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 [2508.13260].

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

The perturbative sector is a central discriminator because many entropy-modified backgrounds can mimic \(\Lambda\)CDM geometrically. In the Bekenstein MHEC case \(n=1\), the entropic component satisfies
\[
\rho_e=\gamma \frac{H^2}{4\pi G},\qquad p_e=-c^2\rho_e,
\]
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 \(\gamma\) enters only through the background functions \(H(a)\) and \(\bar\rho_i(a)\); numerical solutions then follow the \(\Lambda\)CDM growth history within current growth uncertainties. If, instead, the interaction perturbation is neglected, extra damping terms appear and growth is artificially suppressed [2507.08647].

For general \(n\), a spherical top-hat analysis gives the linear matter-contrast equation
\[
\frac{d^2\delta_m}{da^2}
+\frac{3(2-n)}{3-n}\frac{1}{a}\frac{d\delta_m}{da}
-\frac{6n}{(3-n)^2}\frac{\delta_m}{a^2}=0,
\]
with growing mode
\[
\delta_m(z)\propto (1+z)^{\frac{2n}{n-3}}.
\]
In that treatment, \(n>1\) enhances growth relative to \(\Lambda\)CDM, while \(n<1\) suppresses it. The same modified background also affects primordial gravitational waves: \(n>1\) suppresses the relic spectrum, whereas \(n<1\) enhances it [2510.00673].

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 \(n\neq 1\) branch passes a set of cosmographic null tests designed to discriminate it from flat and non-flat \(\Lambda\)CDM [2604.15399].

Early-Universe probes generate a different class of constraints. In generalized-MHR baryogenesis, the entropy
\[
S=\gamma \frac{2n}{n+1}\tilde r_A^{\,n-1}S_{BH}
\]
produces modified background laws
\[
H=\left[\frac{4\pi G(3-n)\rho}{3n\gamma}\right]^{\frac{1}{3-n}},
\qquad
\dot H=-4\pi G\,H^{n-1}n\gamma(\rho+p).
\]
This makes \(\dot R\neq 0\) even during radiation domination and allows the usual gravitational-baryogenesis operator to generate a nonzero baryon asymmetry. For \(T_D=M_I\), the paper finds
\[
0.98\lesssim n<1,
\]
equivalently \(0<1-n\lesssim \mathcal O(10^{-2})\) [2511.01693].

## 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 \(\mathrm{SN}+\mathrm{CC}+\mathrm{BAO}+\mathrm{SH0ES}\) dataset,
\[
H_0=72.28^{+0.90}_{-0.90}\ \mathrm{km\,s^{-1}\,Mpc^{-1}},\qquad
n=0.945^{+0.070}_{-0.070},\qquad
\gamma=1.70^{+0.86}_{-0.67},
\]
with the exact \(\Lambda\)CDM point \((n,\gamma)=(1,1)\) within about \(1\sigma\). Although the raw minimum \(\chi^2\) was comparable to or slightly better than \(\Lambda\)CDM, the Akaike Information Criterion mildly favored \(\Lambda\)CDM in every dataset combination [2508.13260].

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-\(n\) families it found \(\Delta \ln Z\) values between \(2.8524\) and \(2.9806\) relative to \(\Lambda\)CDM, while fixed-coupling scans showed strong preference for weak coupling, roughly
\[
\log_{10}\gamma\lesssim -8.
\]
In that regime the model becomes nearly degenerate with \(\Lambda\)CDM in background parameters, but the combined evidence was reported as slightly to moderately favoring the entropic construction [2512.22103].

Background-only viability analyses are stricter than many early entropic models suggested. In the Cai-Kim formulation built from the generalized MHR with additional \((m,\beta,\alpha)\) 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 \(\Lambda\)CDM-like present-day background [2607.00133].

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
\[
\dot S\ge 0,\qquad \ddot S<0
\]
at late times, indicating approach to a maximum-entropy state, and the fluctuation analysis gives finite horizon-energy variance
\[
\langle(\Delta E)^2\rangle
=
\frac{2\gamma n^2}{(n+1)^2\pi}\frac{\hbar}{G}H^{1-n},
\]
with relative fluctuations decaying as the horizon degrees of freedom grow [2605.12003]. In the emergent-space formulation, the generalized second law was also shown to hold for a universe bounded by the apparent horizon [2512.18054].

Several open issues remain structural rather than numerical. The \(n=3\) branch is exactly \(\Lambda\)CDM in one formulation but singular in later integrated equations. Entropy-first Jacobson constructions imply horizon-area-dependent field equations and quasi-local \(G_{\rm eff}(A)\), 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 [2407.00484][2607.00133].

Source: https://www.emergentmind.com/topics/generalized-mass-to-horizon-entropy-inspired-modified-cosmology