effects of thermal radiation around Kerr black holes from the standpoint of a canonical ensemble. They concluded that slowly rotating black holes (just like Schwarzschild black holes) are unstable in a heat bath, but rapidly rotating holes are less unstable and may even be stable.

\[sec:Stability\] As we pointed out, the standard thermodynamic stability criteria based on the Hessian analysis fails when it is applied to black holes because of the nonadditivity of the entropy function, and also of the impossibility to define extensive thermodynamic quantities (see e.g. [@Arcioni:2004ww]). These problems can be mainly avoided by the Poincaré turning point method [@poincare1885equilibre], which can divide a linear series of equilibrium into subspaces of unstable and less unstable states of equilibria solely from the properties of the equilibrium sequence without solving any eigenvalue equation. The method has been widely applied to problems in astrophysical and gravitating systems, in particular for the study of the thermodynamic stability of black holes in four and also in higher dimensions [@Kaburaki:1993ah; @Katz:1993up; @kaburaki1996critical; @Arcioni:2004ww; @AzregAinou:2012hy]. A well-organized review on the method has been given by Arcioni and Lozano-Tellechea in [@Arcioni:2004ww], which we will mainly follow in the next brief description. Formal proofs of the results 