Generalized Entropic Cosmology Insights

• Generalized entropic cosmology is a framework that replaces the conventional Bekenstein–Hawking area law with generalized entropy forms, leading to modified Friedmann equations and novel dark-energy interpretations.
• The approach employs various entropy functionals—such as Tsallis, Rényi, and logarithmic corrections—to derive entropic-force terms, varying-G effects, and corrections to cosmic dynamics.
• Its applications range from explaining late-time acceleration and early-universe inflation to bounce cosmology, while emphasizing thermodynamic consistency and observational compatibility.

Generalized entropic cosmology denotes a class of cosmological constructions in which the Bekenstein–Hawking area law assigned to the cosmological apparent horizon is replaced by a generalized entropy, and the resulting horizon thermodynamics is used to derive modified Friedmann dynamics, effective dark-energy sectors, entropic-force terms, varying-$G$ realizations, or entropy-monotonicity laws in dynamical spacetimes. In the literature, the central geometric setting is usually a spatially flat FLW or FRW universe with apparent-horizon radius $\tilde r_A=1/H$ and area $A=4\pi \tilde r_A^2$, while the central thermodynamic inputs are the Clausius relation $\delta Q=T\,dS$, the unified first law on the horizon, or closely related relations between horizon entropy, temperature, and energy flow. The subject now encompasses late-time acceleration, effective dark energy, inflation, reheating, bounce cosmology, linear structure growth, primordial gravitational waves, generalized second-law statements, and observational parameter estimation (Nojiri et al., 2022, Khodam-Mohammadi et al., 2023, Luciano et al., 23 Feb 2026, Denkiewicz et al., 26 Dec 2025, Bousso et al., 2015).

1. Thermodynamic setup and scope

The basic premise of entropic cosmology is that the cosmological horizon carries an entropy $S$ and a temperature $T$, and that these quantities enter the cosmological field equations through horizon thermodynamics. Early entropic-force formulations take the Hubble horizon $r_H=c/H$, define an entropic force $F_{\rm ent}=-T\,dS/dr_H$, and interpret the resulting force or pressure as an additional driving term in the acceleration equation [(Komatsu et al., 2013); (Komatsu et al., 2015)]. More recent constructions instead begin directly from the apparent horizon, apply the unified first law $dE=-\delta Q+W\,dV$ or $-dE=T_h\,dS+W\,dV$, and derive modified Friedmann equations in which the entropy deformation is absorbed into an effective fluid or into a deformation of the matter sector (Khodam-Mohammadi et al., 2023, Luciano et al., 23 Feb 2026).

This framework is not restricted to a single entropy functional. The literature includes generalized entropies written directly as functions of $\tilde r_A=1/H$0, two-exponent power laws in the area, logarithmically corrected entropies, generalized mass-to-horizon relations that imply new entropy scalings, and singularity-free entropy functions regular at $\tilde r_A=1/H$1 (Nojiri et al., 2022, Leizerovich et al., 3 Mar 2026, Wu et al., 22 Mar 2026, Odintsov et al., 2022, Denkiewicz et al., 26 Dec 2025). A plausible implication is that “generalized entropic cosmology” is best understood as a research program rather than as a unique model.

A separate but related thermodynamic strand concerns generalized entropy monotonicity in dynamical cosmology. The generalized entropy of a codimension-2 surface is defined as

$\tilde r_A=1/H$2

where $\tilde r_A=1/H$3 is the von Neumann entropy outside the surface and the counterterms render $\tilde r_A=1/H$4 cutoff-independent. On this basis, past and future Q-screens provide a quasi-local setting for a generalized second law in cosmology and black-hole interiors (Bousso et al., 2015).

2. Generalized entropy functionals

A large part of the subject is organized around the choice of generalized entropy. Several representative constructions are recurrent in the literature.

Construction	Entropy or scaling	Representative feature
Four-parameter entropy	$\tilde r_A=1/H$5	Unifies Tsallis, Rényi, Barrow, Sharma–Mittal, Kaniadakis, loop-quantum-gravity limits
Two-exponent entropy	$\tilde r_A=1/H$6	Produces an effective dark-energy sector with two entropic exponents
Log-corrected entropy in GEVAG	$\tilde r_A=1/H$7	Implies an area-dependent $\tilde r_A=1/H$8
Generalized mass-to-horizon relation	$\tilde r_A=1/H$9	Implies $A=4\pi \tilde r_A^2$0 or $A=4\pi \tilde r_A^2$1
Singular-free five-parameter entropy	$A=4\pi \tilde r_A^2$2-regularized $A=4\pi \tilde r_A^2$3	Finite at $A=4\pi \tilde r_A^2$4, suitable for bounce cosmology

The four-parameter entropy introduced by Nojiri and collaborators is explicitly designed to recover familiar nonadditive entropies for suitable parameter choices. In different notational conventions, it appears as the cornerstone of late-time dark-energy models, entropic inflation, reheating, and primordial-gravitational-wave calculations (Nojiri et al., 2022, Odintsov et al., 2023, Odintsov et al., 2023, Odintsov et al., 2024, Adhikary et al., 21 Jul 2025). A closely related four-parameter treatment reconstructs a correction function $A=4\pi \tilde r_A^2$5 by modifying the perfect-fluid energy-momentum tensor as $A=4\pi \tilde r_A^2$6, thereby treating the entropy modification as a deformation of the cosmic fluid sector rather than as an explicit extra component (Khodam-Mohammadi et al., 2023).

A distinct line due to Luciano and Saridakis starts from a microscopic entropy functional with two independent exponents,

$A=4\pi \tilde r_A^2$7

which in an equiprobable ensemble yields

$A=4\pi \tilde r_A^2$8

In the limit $A=4\pi \tilde r_A^2$9 with $\delta Q=T\,dS$0, this reduces to $\delta Q=T\,dS$1, while single-power limits reproduce Tsallis-like behavior (Leizerovich et al., 3 Mar 2026, Luciano et al., 23 Feb 2026).

The generalized mass-to-horizon entropic cosmology (MHEC) adopts a different starting point, namely a generalized scaling between horizon radius and effective mass,

$\delta Q=T\,dS$2

Using $\delta Q=T\,dS$3, one obtains an associated entropy $\delta Q=T\,dS$4 and an entropic pressure. This construction is explicitly presented as thermodynamically consistent because the Hawking temperature is left unmodified while the entropy is induced by the mass-to-horizon ansatz (Denkiewicz et al., 26 Dec 2025, Ali et al., 11 Jul 2025).

The GEVAG framework introduces yet another perspective. There, any modification of the Bekenstein–Hawking law is taken to require an area-dependent Newton coupling in order to preserve the thermodynamic derivation of the Einstein equations and the Bianchi identity. For logarithmic corrections,

$\delta Q=T\,dS$5

so the entropy deformation is reinterpreted as varying-$\delta Q=T\,dS$6 gravity (Wu et al., 22 Mar 2026).

3. Modified Friedmann dynamics and effective sectors

Once the entropy is specified, the common next step is the derivation of modified Friedmann equations. In the two-exponent Luciano–Saridakis construction, the gravity–thermodynamics correspondence on the flat-FRW apparent horizon yields

$\delta Q=T\,dS$7

with

$\delta Q=T\,dS$8

The effective pressure $\delta Q=T\,dS$9 and equation-of-state parameter $S$0 then follow algebraically. In the limiting case $S$1 and $S$2, one recovers $S$3CDM with $S$4 and $S$5 (Leizerovich et al., 3 Mar 2026). In the related 2026 construction based on the same two-exponent area law, the effective dark-energy sector can be quintessence-like, phantom-like, or exhibit phantom-divide crossing, while in all cases $S$6 as $S$7, so the asymptotic future is de Sitter (Luciano et al., 23 Feb 2026).

In the $S$8 approach, the entropy deformation is encoded in a correction to the matter sector. The resulting equations are

$S$9

$T$0

This construction also yields a generalized equation-of-state relation

$T$1

so the usual $T$2 restriction associated with a pure de Sitter interpretation is relaxed (Khodam-Mohammadi et al., 2023).

In MHEC, the entropic sector can be written as an effective fluid with

$T$3

The modified Friedmann system becomes

$T$4

$T$5

A particularly important limit is $T$6, for which $T$7 const and MHEC is exactly $T$8CDM (Denkiewicz et al., 26 Dec 2025). In the Bekenstein case $T$9, one has

$r_H=c/H$0

so the entropic component behaves like a time-varying vacuum energy with $r_H=c/H$1 (Ali et al., 11 Jul 2025).

A broader entropy-area ansatz can also generate a hierarchy of corrections to the Friedmann constraint. If

$r_H=c/H$2

then the generalized Friedmann equation contains a logarithmic $r_H=c/H$3 term, a linear $r_H=c/H$4 term, a volumetric $r_H=c/H$5 term, and higher generalized entropic contributions $r_H=c/H$6 (Chagoya et al., 2024). This suggests that late-time acceleration can emerge from different entropy sectors rather than from a unique entropic deformation.

4. Inflation, reheating, bounce cosmology, and singularity structure

Generalized entropic cosmology has been applied extensively to the early universe. In the four-parameter model of Nojiri, Odintsov, and Paul, the same generalized entropy is used to connect early quasi–de Sitter inflation to late dark energy. The inflationary stage exits at around $r_H=c/H$7 e-folding number, and the inflationary observables are simultaneously compatible with recent Planck data, while the late-time dark-energy equation of state remains consistent with Planck for the same entropy parameters (Nojiri et al., 2022).

A more detailed inflation-to-reheating treatment promotes one entropic parameter to a slowly varying function of e-fold time and treats the entropic energy density as the inflaton. For $r_H=c/H$8, Planck-2018 compatibility requires $r_H=c/H$9, $F_{\rm ent}=-T\,dS/dr_H$0, and $F_{\rm ent}=-T\,dS/dr_H$1; analogous ranges are reported for $F_{\rm ent}=-T\,dS/dr_H$2 and $F_{\rm ent}=-T\,dS/dr_H$3. Reheating is modeled by a constant $F_{\rm ent}=-T\,dS/dr_H$4, the Hubble rate evolves continuously from quasi–de Sitter to power law, and instantaneous reheating is possible for $F_{\rm ent}=-T\,dS/dr_H$5 (Odintsov et al., 2023). In a related scalar-field realization, the minimally coupled quadratic potential $F_{\rm ent}=-T\,dS/dr_H$6, which is ruled out in standard scalar-field cosmology, becomes viable once the apparent-horizon entropy is generalized in the four-parameter form (Odintsov et al., 2023).

Primordial gravitational waves provide an additional early-universe probe. In the four-parameter horizon-cosmology model, the present-day spectrum acquires a scale-invariant plateau for modes re-entering during radiation and a blue-tilted reheating tail with

$F_{\rm ent}=-T\,dS/dr_H$7

The paper emphasizes that the plateau amplitude, the break frequency $F_{\rm ent}=-T\,dS/dr_H$8, and the high-frequency tilt can in principle be inverted to recover the entropic parameters, with sensitivity forecasts discussed for SKA, LISA, DECIGO, and BBO (Odintsov et al., 2024).

Bounce cosmology is addressed by a singularity-free five-parameter entropy that remains finite at $F_{\rm ent}=-T\,dS/dr_H$9 because it regularizes the Bekenstein–Hawking dependence through $dE=-\delta Q+W\,dV$0. This framework admits symmetric exponential and quasi-matter bounces; for the quasi-matter bounce, perturbations originate in the deep sub-Hubble regime far before the bounce, and the example $dE=-\delta Q+W\,dV$1, $dE=-\delta Q+W\,dV$2 gives $dE=-\delta Q+W\,dV$3 and $dE=-\delta Q+W\,dV$4, in agreement with Planck 2018 (Odintsov et al., 2022).

The GEVAG treatment of logarithmic corrections produces two distinct ultraviolet behaviors depending on the sign of the logarithmic coefficient. For $dE=-\delta Q+W\,dV$5, $dE=-\delta Q+W\,dV$6 in the ultraviolet, the friction term in slow roll is effectively doubled, and slow-roll inflation becomes more natural. For $dE=-\delta Q+W\,dV$7, one finds a maximum density, $dE=-\delta Q+W\,dV$8, and the big-bang singularity is replaced by density saturation, while the GEVAG dynamics avoids the sudden singularity that appears in the constant-$dE=-\delta Q+W\,dV$9 approach with the same logarithmic correction (Wu et al., 22 Mar 2026). A late-time singularity analysis in a related generalized entropic cosmology with a viscous logarithmic dark fluid further reports that Hawking-radiation back-reaction can soften or completely remove a would-be Big Rip (Elizalde et al., 4 May 2026).

5. Structure growth and observational constraints

Observational work on generalized entropic cosmology is now diverse in both data selection and parameterization. The first observational confrontation of the Luciano–Saridakis two-exponent model, restricted to $-dE=T_h\,dS+W\,dV$0, uses Cosmic Chronometers, Pantheon$-dE=T_h\,dS+W\,dV$1 Type Ia supernovae calibrated with SH0ES, DESI DR2 BAO, and compressed Planck 2018 CMB information. The reported $-dE=T_h\,dS+W\,dV$2 constraints are

$-dE=T_h\,dS+W\,dV$3

$-dE=T_h\,dS+W\,dV$4

with derived $-dE=T_h\,dS+W\,dV$5. In the restricted parameter space considered, the $-dE=T_h\,dS+W\,dV$6CDM point $-dE=T_h\,dS+W\,dV$7 lies outside the joint $-dE=T_h\,dS+W\,dV$8 region, and the model gives $-dE=T_h\,dS+W\,dV$9, simultaneously consistent with Pantheon$\tilde r_A=1/H$00+SH0ES and the Planck shift parameters (Leizerovich et al., 3 Mar 2026).

A four-parameter generalized-entropic dark-energy model confronted with CC, PantheonPlus+SH0ES, DESI DR1, and compressed Planck likelihoods also returns $\tilde r_A=1/H$01 near $\tilde r_A=1/H$02. For instance, the DESI+Planck+Pantheon fit gives

$\tilde r_A=1/H$03

while the CC+Pantheon+SH0ES fit gives

$\tilde r_A=1/H$04

The same work reports $\tilde r_A=1/H$05 and $\tilde r_A=1/H$06 for CC alone relative to $\tilde r_A=1/H$07CDM, and mild tension but comparable performance on larger data combinations (Adhikary et al., 21 Jul 2025).

The MHEC program extends the comparison beyond background probes. A joint likelihood using Pantheon$\tilde r_A=1/H$08 with SH0ES calibration, DESI DR2 BAO, compressed CMB distance priors, WiggleZ BAO+RSD, SDSS-IV DR14 QSO BAO+RSD, and uncorrelated $\tilde r_A=1/H$09 points reports moderate Bayesian evidence in favor of weak-coupling MHEC. For fixed $\tilde r_A=1/H$10, $\tilde r_A=1/H$11 gives $\tilde r_A=1/H$12 relative to $\tilde r_A=1/H$13CDM, whereas $\tilde r_A=1/H$14 is strongly disfavored with $\tilde r_A=1/H$15 (Denkiewicz et al., 26 Dec 2025). In the Bekenstein case $\tilde r_A=1/H$16, a separate perturbative analysis emphasizes the distinction between a fully perturbed interaction term and the approximation that neglects its perturbation. In the fully perturbed case, the $\tilde r_A=1/H$17 source cancels exactly at first order, the matter-growth history follows $\tilde r_A=1/H$18CDM within current growth uncertainties, and $\tilde r_A=1/H$19 $\tilde r_A=1/H$20 data points give a global minimum at $\tilde r_A=1/H$21, $\tilde r_A=1/H$22, while $\tilde r_A=1/H$23 is statistically indistinguishable (Ali et al., 11 Jul 2025).

A more agnostic generalized-Friedmann approach based on logarithmic, linear, and volumetric entropy corrections has also been constrained with Hubble-parameter measurements, BAO, and strong-lensing systems. In the quartic model, the best-fit values from OHD+SLS are

$\tilde r_A=1/H$24

with $\tilde r_A=1/H$25, compared with $\tilde r_A=1/H$26 for $\tilde r_A=1/H$27CDM on the same data. In that fit, the logarithmic correction is small, while the volumetric term is dominant in driving acceleration (Chagoya et al., 2024).

6. Thermodynamic consistency, generalized second law, and recurrent controversies

Several papers make thermodynamic consistency itself the primary issue. In the single-power entropic-force model $\tilde r_A=1/H$28, preserving the Legendre structure requires the temperature to scale as

$\tilde r_A=1/H$29

so for $\tilde r_A=1/H$30 the temperature differs from the Hawking one. The same analysis finds that the pure single-power model yields a constant deceleration parameter and cannot by itself reproduce both a decelerated matter era and a later acceleration, although it fits Pantheon supernova data well with $\tilde r_A=1/H$31 from $\tilde r_A=1/H$32 and $\tilde r_A=1/H$33 from $\tilde r_A=1/H$34 (Zamora et al., 2022). This directly contradicts the widespread assumption that generalized entropy can always be combined with the standard Hawking temperature without further consistency conditions.

A closely related concern is whether a chosen entropy law is compatible simultaneously with the unified first law and the standard Friedmann equation. For the Tsallis–Cirto volume law $\tilde r_A=1/H$35, it was shown that the entropy alone cannot satisfy both unless the effects of a dark-energy candidate on the horizon entropy are included; in that setting, the same entropy law can be used to connect Verlinde’s entropic-force picture with Padmanabhan’s emergent-space proposal (Moradpour, 2016). Likewise, in a phenomenological model with $\tilde r_A=1/H$36, the continuity equation derived from the first law coincides exactly with the one derived from the Friedmann and acceleration equations only for $\tilde r_A=1/H$37, namely the Bekenstein area law; for $\tilde r_A=1/H$38 the two differ by a factor $\tilde r_A=1/H$39, and for $\tilde r_A=1/H$40 there is no consistency (Komatsu et al., 2015).

Against this background, later papers explicitly advertise thermodynamic consistency as a design criterion. MHEC states that the Clausius relation $\tilde r_A=1/H$41 holds exactly for any $\tilde r_A=1/H$42, that the unmodified Hawking temperature is retained, and that the framework avoids the inconsistencies of earlier entropy-force models (Ali et al., 11 Jul 2025). GEVAG similarly insists that modifying the entropy requires an accompanying varying-$\tilde r_A=1/H$43 sector in order to preserve the thermodynamic derivation of the field equations and the Bianchi identity (Wu et al., 22 Mar 2026).

The generalized second law provides an additional consistency test. In the Q-screen formulation, a regular past Q-screen with foliation $\tilde r_A=1/H$44 satisfies

$\tilde r_A=1/H$45

provided the spacetime obeys the Quantum Focussing Conjecture. In a radiation-dominated flat FRW example, one finds

$\tilde r_A=1/H$46

which was presented as the first local or quasi-local second-law statement holding in fully dynamical cosmological spacetimes without teleological event-horizon constructions (Bousso et al., 2015). Other generalized-entropic models also check the GSL directly. In the $\tilde r_A=1/H$47 framework, convexity requires $\tilde r_A=1/H$48 and $\tilde r_A=1/H$49, with $\tilde r_A=1/H$50 implying $\tilde r_A=1/H$51 (Khodam-Mohammadi et al., 2023). In GEVAG, the total entropy production satisfies

$\tilde r_A=1/H$52

and the paper identifies the parameter ranges in which the GSL remains valid during expansion, inflation, or bounce phases (Wu et al., 22 Mar 2026).

Open directions are stated explicitly across the literature. They include a direct independent proof of the Quantum Focussing Conjecture, a microscopic origin for $\tilde r_A=1/H$53 or other generalized horizon entropies in cosmology, perturbation-level implementations with full CMB and LSS likelihoods, exploration of the full $\tilde r_A=1/H$54 parameter plane, and a quantum-gravity derivation of the generalized entropic couplings $\tilde r_A=1/H$55, $\tilde r_A=1/H$56, $\tilde r_A=1/H$57, $\tilde r_A=1/H$58, and related parameters (Bousso et al., 2015, Leizerovich et al., 3 Mar 2026, Chagoya et al., 2024). A plausible implication is that the long-term viability of generalized entropic cosmology will depend less on background-level fits alone than on thermodynamic self-consistency, perturbation theory, and microphysical interpretation.