Black Hole Solution in f(R,G) Gravitational Theory Coupled with Scalar Field

In this work, we explore a class of spherically symmetric black hole (BH) solutions within the framework of modified gravity, focusing on a non-ghost-free f(R,G) theory coupled to a scalar field. We present a novel black hole geometry that arises as a deformation of the Schwarzschild solution and analyze its physical and thermodynamic properties. Our results show that the model satisfies stability conditions, with the Ricci scalar R, as well as its first and second derivatives, remaining positive throughout the spacetime. The solution admits multiple horizons and exhibits strong curvature singularities compared to those in general relativity. Furthermore, it supports a non-trivial scalar field potential. A comprehensive thermodynamic analysis is performed, including evaluations of the entropy, temperature, heat capacity, and quasi-local energy. We find that the black hole exhibits thermodynamic stability within certain ranges of model parameters. In addition, we investigate geodesic deviation and derive the conditions necessary for stability within the f(R,G) gravitational framework.