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 ofSchwarzschild black holes under Planck-scale modified kinematics, using a cubic entropy correction derived from a well-known phenomenological MDR with leading correction η E³/EP η E 3 / E P . Enforcing physical constraints ( S'(rₕ) > 0 S ′ ( r h ) > 0 , $$T > 0$$ T > 0 ) via the entropy-geometry correspondence, we find a single unstable branch with $$w = -1$$ w = - 1 and $$W = -1$$ W = - 1 for both signs of the correction parameter. A second root suggesting stability ( $$w = +1$$ 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.