Conformal Invariance and Phase Transitions: Implications for Stable Black Hole Horizons?

The behavior of black hole horizons under extreme conditions—such as near collapse or phase transitions—remains less understood, particularly in the context of soft hair and Aretakis instabilities. We show that the breakdown of conformal symmetry during the balding phase induces a topological reorganization of the horizon, leading to divergent entropy corrections and emergent pressure terms. These corrections exhibit universal scaling laws, analogous to quantum phase transitions in condensed matter systems, with extremal limits functioning as quantum critical points. Interestingly, by employing quasi-equilibrium boundary conditions, one could stabilize horizon dynamics without explicitly introducing ad hoc higher-order corrections, further limiting the universal applicability of conformal invariance in black hole physics.