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Spontaneous scalarization of black holes in scalar-Gauss-Bonnet (sGB) gravityis a very interesting phenomenon allowing black holes to circumvent the no-hairtheorem and acquire scalar hair while leaving the weak field regime of thetheory practically unaltered. It was recently shown that if we allow for adifferent form of the coupling between the scalar field and the Gauss-Bonnetinvariant, a new type of scalarization is possible beyond the standard one thatcan be excited only nonlinearly. In this case the spectrum of black holesolutions can be more complicated, naturally opening the question about theirstability. The goal of the present paper is to study this question includingthe possible loss of hyperbolicity of the radial perturbation equation. We showthat one of the nonlinearly scalarized phases of the Schwarzschild black holecan be stable and hyperbolic for large enough black hole masses. These resultsestablish the studied hairy black holes as viable astrophysical candidates.

Radial perturbations of scalar-Gauss-Bonnet black holes beyond spontaneous scalarization