  This study is aimed at providing a thorough analysis of the classical
thermodynamic stability of black holes within the Anti-de Sitter (AdS) family.
We utilize the Nambu bracket formalism to calculate local heat capacities and
employ Sylvester's criterion in the mass-energy ensemble to determine both
local and global thermodynamic stability regions. Emphasizing the crucial role
of the cosmological constant, we establish the conditions necessary for the
existence of thermodynamically stable black hole configurations. Our work
highlights the applicability of classical thermodynamics in understanding black
hole physics, while acknowledging the potential deviations that may impact the
observables of the system beyond the classical level.
