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In this paper, we study static and spherically symmetric black hole (BH)solutions in the scalar-tensor theories with the coupling of the scalar fieldto the Gauss-Bonnet (GB) term $\xi (\phi) R_{\rm GB}$, where $R_{\rmGB}:=R^2-4R^{\alpha\beta}R_{\alpha\beta}+R^{\alpha\beta\mu\nu}R_{\alpha\beta\mu\nu}$is the GB invariant and $\xi(\phi)$ is a function of the scalar field $\phi$.Recently, it was shown that in these theories scalarized static and sphericallysymmetric BH solutions which are different from the Schwarzschild solution andpossess the nontrivial profiles of the scalar field can be realized for certainchoices of the coupling functions and parameters. These scalarized BH solutionsare classified in terms of the number of nodes of the scalar field. It was thenpointed out that in the case of the pure quadratic order coupling to the GBterm, $\xi(\phi)=\eta \phi^2/8$, scalarized BH solutions with any number ofnodes are unstable against the radial perturbation. In order to see how ahigher order power of $\phi$ in the coupling function $\xi(\phi)$ affects theproperties of the scalarized BHs and their stability, we investigate scalarizedBH solutions in the presence of the quartic order term in the GB couplingfunction, $\xi(\phi)=\eta \phi^2 (1+\alpha \phi^2)/8$. We clarify that theexistence of the higher order term in the coupling function can realizescalarized BHs with zero nodes of the scalar field which are stable against theradial perturbation.

Scalarized black holes in the presence of the coupling to Gauss-Bonnet gravity