What topology reveals about regular black hole stability

Using a topological method based on winding numbers of a vector field derived from free energy, the authors map out the stability and phase transitions of a family of regular (singularity-free) black holes that includes Hayward and Bardeen geometries as special cases. Regular configurations always carry two topological defects with opposite winding numbers, giving a net charge of zero — unlike the Schwarzschild black hole, which has a single unstable branch. The result ties the presence of regularization parameters directly to observable changes in thermodynamic phase structure.