Thermodynamics in $f(R,\mathcal{L})$ theories: Apparent horizon in the FLRW spacetime

In this paper we study the recently proposed $f(R,\mathcal{L})$ theories from a thermodynamic point of view. The uniqueness of these theories lies in the fact that the space-time curvature is coupled to the baryonic matter instead of exotic matter (in the form of scalar field). We investigate the viability of these theories from the point of view of the thermodynamic stability of the models. To be more precise here we are concerned with the thermodynamics of the apparent horizon of Friedmann-Lemaitre-Robertson-Walker (FLRW) spacetime in the background of the $f(R,\mathcal{L})$ theory. We consider several models of $f(R,\mathcal{L})$ theories where both minimal and non-minimal coupling has been considered. Various thermodynamic quantities like entropy, enthalpy, internal energy, Gibbs free energy, etc. are computed and using their allowed ranges various model parameters are constrained.