thermodynamics of extensive systems, the local thermodynamic stability (defined as the Hessian of the entropy has no positive eigenvalues) is linked to the dynamical stability of the system. This stability criteria, however, strongly relies on the additivity of the entropy function, which is a property that clearly does not hold for black holes. The simplest example of this discrepancy is the Schwarzschild black hole which is known to be perturbatively stable but has a negative specific heat (positive Hessian). Black hole phase transitions are also strongly related to the stability properties of the system (in particular the Hawking–Page transition), and since the standard methods are not reliable in nonextensive thermodynamics, one has to be very careful when considering stability and phase transitions in strongly gravitating systems.

Avoiding the complications arising from the Hessian approach to the stability problem of black holes, an alternative technique was proposed in a series of paper by Kaburaki et al. [@Kaburaki:1993ah; @Katz:1993up; @kaburaki1996critical]. In these works the so-called “Poincaré turning point method of stability” [@poincare1885equilibre] has been applied to the problem, which is a topological approach and does not depend on the additivity of the entropy function. More recently this method has also been used 