Analytically approximated scalarized black holes and their thermodynamic stability

It is recently shown that, besides the Schwarzshcild black hole solution, there exist also scalarized black hole solutions in some Einstein-scalar-Gauss-Bonnet theories. In this paper, we construct analytical expressions for the metric functions and scalar field configurations for these scalarized black hole solutions approximately by employing the continued fraction parametrization method and investigate their thermodynamic stability. It is found that the horizon entropy of a scalarized black hole is always smaller than that of a Schwarzschild black hole, which indicates that these scalarized black holes may decay to Schwarzschild black holes by emission of scalar waves. This fact also implies the possibility to extract the energy of scalar charges.