Topological black holes with curvature induced scalarization in the extended scalar-tensor theories

We study the stability of topological black holes in the presence of a cosmological constant and a scalar field coupled to the Gauss-Bonnet (GB) term in the extended scalar-tensor theories. We find two competing effects. As the strength of the coupling λ of the scalar field to the GB term is increasing, the matter is interacting more strongly with gravity while as the hyperbolicity ζ of spacetime is getting larger the kinetic effects of matter tend to dominate. Calculating both analytically and numerically the quasinormal modes (QNMs) we found for each λ a critical value of ζ, below which there is an instability. When the coupling constant λ is getting very large, all of the QNMs develop a positive imaginary part indicating an instability. This behavior indicates a phase transition to a scalarized topological black hole induced by curvature effects.