Apparent horizon thermodynamics in an exponential $f(Q)$ gravity model

Abstract

We investigate the thermodynamics of the apparent horizon in an exponential $f(Q)$ gravity model characterized by the two parameters $b$ and $n$, within a spatially flat Friedmann--Lemaître--Robertson--Walker background and the coincident gauge. Focusing on the $n=1$ solution, we use the approximate cosmological solution for the Hubble parameter up to second order in the exponential parameter $b$ to study the redshift evolution of the apparent-horizon radius and the Kodama--Hayward temperature. We formulate Hayward&#39;s unified first law in terms of the Misner--Sharp--Hernandez mass, the work density, and the energy-supply vector, and show that the horizon dynamics admits an equilibrium thermodynamic description. The associated entropy differential is proportional to $f_Q+2Qf_{QQ}$, yielding exponentially suppressed corrections to the Bekenstein--Hawking area law and recovering $S=A/4$ in the limit $b\to0$. We then examine the generalized second law (GSL) by including the entropy of matter inside the apparent horizon. When the matter and horizon temperatures are identified, we adopt the GSL viability criterion $f_Q+2Qf_{QQ}\geq0$ previously derived in the literature. The observationally motivated best-fit values of $b$ satisfy this condition over the redshift interval studied and produce departures from $Λ$CDM mainly at late times. In contrast, sufficiently large positive values, approximately $b>0.26$, can violate the GSL in the future region $z<0$. These results show that apparent-horizon thermodynamics provides a complementary constraint on the parameter space of exponential $f(Q)$ cosmology.