The generalized second law as a thermodynamic selection criterion for dynamical dark energy

Abstract

The growing number of generalized horizon entropy proposals has led to a wide variety of modified cosmological models, yet there is currently no general physical principle capable of discriminating among them. We show that the generalized second law (GSL) of thermodynamics provides such a criterion. Considering a general modified Friedmann framework described by an arbitrary function $f(H)$ of the Hubble parameter, and allowing the apparent horizon and the cosmic fluid to evolve out of thermal equilibrium, we derive model-independent constraints on the asymptotic scaling of the horizon entropy, $S_A\propto A^k$. We find that phantom evolution requires $k\ge(3ω-1)/(2ω)$, with $ω$ the total fluid equation of state, whereas quintessence imposes the complementary upper bound. Additionally, in the exact thermal equilibrium limit, the dynamical coupling strictly enforces $k \le 2$, recovering the non-equilibrium quintessence bound. The two bounds converge to the unique value $k=2$ as the phantom divide is approached, indicating that a smooth crossing of the phantom divide is thermodynamically associated with a quadratic entropy-area scaling, where the effective theory degenerates into a logarithmic gravity framework. Applying this criterion to representative generalized entropy models shows that many commonly used proposals are constrained by the generalized second law, whereas multiparameter constructions are naturally compatible with the required asymptotic behavior. Finally, we demonstrate that the same thermodynamic selection principle extends to derivative-dependent cosmologies described by $f(H,\dot H)$, highlighting its robustness beyond entropy functionals that depend only on the horizon area.