Thermodynamic stability of pure black holes

The stability limits in a linear series of charged rotating black holes are found, assuming that the spectrum of eigenvalues of the matrix of second-order derivatives in the partition function is not degenerate. Among eight different ensembles, the four ensembles with mass-energy as control parameter are stable with respect to arbitrary, not necessarily small, axially symmetric perturbations. The results are valid when the radiation energy is a negligible fraction of the total energy and therefore the backreaction on the metric can be neglected as well. Three ensembles in a heat bath are unstable when the total mass is high compared to angular momentum and charge. They are less unstable and may be stable if the mass is low enough. Black holes with fixed temperature, angular velocity and electric potential are unstable. Also considered are the stability limits of 'inner-horizon thermodynamic' systems. The four ensembles with given mass are either stable or unstable. The same is true for inner horizons in a heat bath with fixed angular momentum and fixed charge or electric potential. Inner horizons in a heat bath at constant angular velocity and fixed charge or fixed electric potential are less unstable and may be stable with relatively high (rather than low) mass-energy. Signs of thermodynamic coefficients in gravitating systems are not indicators of stability. However, changes of signs and changes of stability are correlated. In the canonical ensemble heat capacities turn from + infinity in stable systems to - infinity in unstable system. Moments of inertia and electric capacities or their negative inverses behave in similar ways.