Generalized Second Law of Thermodynamics in $f(Q)$ Symmetric Teleparallel Gravity

Abstract

We investigate the validity of the generalized second law (GSL) of thermodynamics on the cosmological apparent horizon in $f(Q)$ symmetric teleparallel gravity, using the Wald-type entropy prefactor $A{f}_Q/4G$ together with a correction term derived directly from the modified Friedmann equations. We derive, and verify by explicit symbolic computation, a closed-form expression for the apparent horizon entropy in $f(Q)$ gravity and show that its time derivative combines with the matter entropy rate into a remarkably compact master formula, $T_A{\dot{S}}_{\mathrm{tot}}=4\pi \dot{H}(\dot{H}{f}_Q+H{\dot{f}}_Q)/H^4$ (in units $8\pi G=1$), which correctly reduces to the known general-relativistic limit. We apply this formalism to three observationally viable $f(Q)$ models --- power-law, exponential, and logarithmic. We find that the GSL is satisfied throughout cosmic history for all three models, and relate this robustness to a general, theory-independent structural condition: for ordinary pressureless matter obeying the null energy condition, GSL validity reduces to the sign of $\dot{H}$, i.e., to the absence of a phantom crossing in the total effective expansion history. The geometric sector exhibits model-dependent energy conditions, power-law and exponential models violate only the strong condition, while the logarithmic model violates the null, weak, and dominant conditions for \(z>-1\), and the strong condition for \(z<3.84\).