Spherically symmetric black holes in gravity from entropy and spontaneous emission

We investigate static and dynamical spherically symmetric black hole solutions within the Gravity from Entropy (GfE) framework. We derive and solve the modified vacuum field equations for a static, spherically symmetric spacetime, revealing that the classical Schwarzschild geometry receives perturbative corrections scaling as $r^{-4}$. We establish that the GfE framework is consistent with current strong-field astrophysical observations. Higher-order geometric stresses inherent to the GfE vacuum drive a consistent mass-evolution profile. In the limit of large black hole mass, the theory predicts a constant background evaporation rate $ -β/24$, suggesting an inherent "entropic leakage" of the vacuum. At intermediate scales, the framework replicates the standard Hawking radiation mass-loss law as $\dot{M} \propto M^{-2}$ through a purely classical response of the modified background.