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It has been shown that the synergy of a scalar field coupling with both theRicci scalar and the Gauss-Bonnet invariant significantly affects theproperties of scalarized black holes and neutron stars, including their domainof existence and the amount of scalar hair they carry. Here we study the radialstability of scalarized black-hole solutions. We demonstrate that they arestable against radial perturbations for Ricci couplings consistent with both alate-time cosmological attractor and the evasion of binary pulsar constraints.In addition, we investigate the effect of the Ricci coupling on thehyperbolicity of the equation governing linear, radial perturbations and showthat it significantly reduces the region over which hyperbolicity is lost.

Stable spontaneously-scalarized black holes in generalized scalar-tensor theories