Gravity theory of the generalized mass to horizon entropy The Iyer Wald approach

In this letter, our aim is to study the gravitational origin of the $\mathit{\text{generalized}}\mathit{\text{mass-to-horizon}}\mathit{\text{entropy}}$ (GMHE), described by an entropic exponent index $n$ and a multiplicative parameter $\gamma$. We employ Iyer-Wald’s approach within the modified $\mathfrak{f}(\mathfrak{R})$ gravity scenario in order to assess the horizon entropy of the black hole (BH) by considering solutions with constant curvature for the spherically symmetric vacuum field equations. In this process, we explicitly show that the GMHE can effectively be reconstructed from a generalized Lagrangian $\mathfrak{L}\propto {\mathfrak{R}}^{1+\epsilon }$, where $\epsilon \equiv \frac{n-1}{2}$ measures tiny departures from the standard formulation of general relativity (GR). Finally, we discuss the physical implications of our results in relation to cosmology and the prospects for alleviating the thermodynamic instability of Schwarzschild BHs.