  Scalar-tensor theories of gravity where a new scalar degree of freedom
couples to the Gauss-Bonnet invariant can exhibit the phenomenon of spontaneous
black hole scalarization. These theories admit both the classic black holes
predicted by general relativity as well as novel hairy black hole solutions.
The stability of the hairy black holes is strongly dependent on the precise
form of the scalar-gravity coupling. A radial stability investigation revealed
that all scalarized black hole solutions are unstable when the coupling between
the scalar field and the Gauss-Bonnet invariant is quadratic in the scalar,
whereas stable solutions exist for exponential couplings. Here we elucidate
this behavior. We demonstrate that, while the quadratic term controls the onset
of the tachyonic instability that gives rise to the black hole hair, the
higher-order coupling terms control the nonlinearities that quench that
instability, and hence also control the stability of the hairy black hole
solutions.
