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We extend the analysis, initiated in arXiv:1901.01269, of the thermodynamicstability of four-dimensional asymptotically flat hairy black holes byconsidering a general class of exact solutions in Einstein-Maxwell-dilatontheory with a non-trivial dilaton potential. We find that, regardless of thevalues of the parameters of the theory, there always exists a sub-class ofhairy black holes that are thermodynamically stable and have the extremal limitwell defined. This generic feature that makes the equilibrium configurationslocally stable should be related to the properties of the dilaton potentialthat is decaying towards the spatial infinity, but behaves as a box close tothe horizon. We prove that these thermodynamically stable solutions are alsodynamically stable under spherically symmetric perturbations.

Thermodynamically stable asymptotically flat hairy black holes with a dilaton potential: the general case