We revisit the classical thermodynamic stability of the standard black hole solutions by implementing the intrinsic necessary and sufficient conditions for stable global and local thermodynamic equilibrium. The criteria for such equilibria are quite generic and well-established in classical thermodynamics, but they have not been fully utilized in black hole physics. We show how weaker or incomplete conditions could lead to misleading or incorrect results for the thermodynamic stability of the system. We also stress the importance of finding all possible local heat capacities in order to fully describe the classical equilibrium picture of black holes. Finally, we thoroughly investigate the critical and phase transition curves and the limits of the classical analysis. This paper is the first in the line of intended works on thermodynamic stability of black holes in modified theories of gravity and holography.

We suggest a finite-time geometric optimization framework to analyze thermal fluctuations and optimal processes in black holes. Our approach implement geodesics in thermodynamic space to define optimal pathways between equilibrium and non-equilibrium states. Since thermodynamic metrics need not be positive-definite, the method ensures a positive thermodynamic length by incorporating simple scale factor into the metric. We show that the thermodynamic length and the scale factor are sensitive to phase transitions in entropy representation, addressing a key gap in Hessian thermodynamic geometry. Additionally, we link the scale factor to the sign of thermodynamic curvature, connecting it to the information geometry governing optimal processes. Our results indicate that optimal fluctuations can drive the evaporation of Schwarzschild and Kerr black holes, which may significantly differ from Hawking radiation. We also explore optimal accretion-driven processes supported by an external inflow of energy.

In this work we present a brief introduction to the concepts of classical global and local thermodynamic stability from thermodynamic information geometry perspective. We state the necessary and sufficient conditions for stable global thermodynamic equilibrium. Novel relations between Hessians, defined on different thermodynamic manifolds, have been derived, which lead to new expressions for the Weinhold and Ruppeiner thermodynamic metrics. We show that our approach is equivalent to the Nambu bracket formalism traditionally used in the calculation of the heat capacities of the system. Finally, we make our remarks on the concept of local thermodynamic stability and its relation to the global one.

We investigate the geometric Nielsen complexity of the unitary evolution of the supersymmetric quantum harmonic oscillator (SQHO), formulated on the Heisenberg–Weyl Lie superalgebra \(\mathfrak{s}\mathfrak{h}(\left. 2 \right|2)\).