  Spontaneous scalarization of black holes in scalar-Gauss-Bonnet (sGB) gravity
is a very interesting phenomenon allowing black holes to circumvent the no-hair
theorem and acquire scalar hair while leaving the weak field regime of the
theory practically unaltered. It was recently shown that if we allow for a
different form of the coupling between the scalar field and the Gauss-Bonnet
invariant, a new type of scalarization is possible beyond the standard one that
can be excited only nonlinearly. In this case the spectrum of black hole
solutions can be more complicated, naturally opening the question about their
stability. The goal of the present paper is to study this question including
the possible loss of hyperbolicity of the radial perturbation equation. We show
that one of the nonlinearly scalarized phases of the Schwarzschild black hole
can be stable and hyperbolic for large enough black hole masses. These results
establish the studied hairy black holes as viable astrophysical candidates.
