A winding number analysis of Schwarzschild black hole stability in light of Planck-scale modified kinematics

Determining whether Planck-scale effects can stabilize black holes addresses fundamental questions about black hole evaporation and quantum gravity consistency. Here, we analyze the thermodynamic topology of Schwarzschild black holes under Planck-scale modified kinematics, using a cubic entropy correction derived from a well-known phenomenological MDR with leading correction $\eta E^3\mathrm{/}{E}_P$. Enforcing physical constraints ($S^{\mathrm{\prime }}(r_h)>0$, $T>0$) via the entropy-geometry correspondence, we find a single unstable branch with $w=-1$ and $W=-1$ for both signs of the correction parameter. A second root suggesting stability ($w=+1$) is excluded due to negative mass/temperature and lies outside the perturbative regime. Thus, this class of MDRs does not yield stable Schwarzschild black holes. However, MDRs with different leading-order corrections may behave otherwise, leaving the search for Planck-scale stabilization an open endeavor.