Scalarization of charged Taub–NUT black hole and the entropy bound

Abstract We investigate the spontaneous scalarization of charged Taub–NUT black holes within the framework of Einstein–Maxwell-scalar-Gauss–Bonnet gravity. By selecting a suitable coupling function, the theory admits the analytic charged Taub–NUT geometry as a solution. We demonstrate that this scalar-free background becomes unstable within specific parameter regimes, leading to the bifurcation of a new branch of hairy charged Taub–NUT black holes. These solutions are characterized by a two-dimensional parameter space spanned by the electric charge and the NUT parameter. We conduct a systematic study of their properties, specifically the scalar charge, temperature, and entropy. Our analysis reveals that the entropy of the scalarized solutions exhibits particularly compelling features. Two characteristics emerge: first, the entropy of the hairy black hole is strictly greater than that of its scalar-free counterpart; second, the entropy reaches a local maximum precisely at the bifurcation point. Notably, when the electric charge is held fixed, this maximum entropy value remains constant over a certain range of the mass parameter.