Abstract In this study, we investigate rotating black hole solutions within a scalar-Gauss–Bonnet gravity framework that incorporates a squared Gauss–Bonnet term. By employing a quadratic–exponential coupling function between the scalar field and the Gauss–Bonnet invariant, we derive both the standard General Relativity solutions and novel scalarized black hole configurations. Utilizing a pseudo-spectral method to solve the coupled field equations, we examine how black hole spin and coupling constants influence the existence and properties of these solutions. Our findings reveal that both the rotation of the black hole and the squared coupling term effectively constrain the parameter space available for scalarization. Moreover, we demonstrate that, over a wide range of parameters, scalarized black holes exhibit higher entropy than Kerr black holes of equivalent mass and spin, indicating that they are thermodynamically favored. These results significantly expand the phase space of black holes in modified gravity theories.
Liu et al. (Sat,) studied this question.