Black hole zeroth law of the Einstein-complex scalar-Gauss-Bonnet gravity

The zeroth law of black hole mechanics asserts that the stationary black hole has a constant surface gravity on the event horizon. This study investigates the zeroth law within the Einstein-Scalar-Gauss-Bonnet gravity, particularly involving a complex scalar field (ECSGB). Assuming that the Lie derivative of the complex scalar field is proportional to itself other than stationary, and that ECSGB gravity converges to the Einstein's gravity analyticaly as the coupling constant approaches zero, as well as the minimally coupled matter fields satisfy the dominant energy condition, we demonstrate that the surface gravity remains constant over the entire event horizon without reliance on additional symmetries.