Critical slowing down of black hole phase transition and universal dynamic scaling in AdS black holes

We investigate the dynamical critical behaviour of black hole phase transitions in anti de Sitter spacetime by extending the stochastic framework of free energy landscape dynamics to Kerr AdS black holes. By analyzing the Langevin evolution of the entropy (in contrast to the horizon radius in the RN-AdS case) near criticality, we demonstrate that the system exhibits pronounced critical slowing down, characterized by a significant increase of the autocorrelation time as the critical point is approached. This behaviour is further confirmed by the lowest eigenvalue of the Fokker-Planck equation. By analysing the dynamics along different thermodynamic paths, including variations in temperature, pressure, and angular momentum, and considering both directions - towards and away from criticality, we find that the relaxation time obeys a robust scaling relation, $\tau=|\epsilon|^{-2/3}$ near criticality. The same scaling exponent is obtained for RN-AdS, Kerr-AdS, and Bardeen black holes, suggesting the existence of an underlying universal dynamical behaviour across distinct black hole systems. Our results establish a connection between the geometry of the free energy landscape, stochastic nonequilibrium dynamics, and universal critical phenomena in black hole thermodynamics.