  We examine the thermodynamic characteristics and phase structures of a black
hole, where the black hole horizon could be a hypersurface with positive, zero,
or negative constant curvature, within the framework of Einstein-Maxwell
theory, incorporating a negative cosmological constant and a Gauss-Bonnet
correction. Our research follows the topological approach to black hole
thermodynamics where we treat anti-de Sitter (AdS) black holes as topological
defects in thermodynamic space. We study the nature of the black hole's
critical points and local stability by computing the winding
numbers/topological charge associated with the zero point of the vector field,
derived from the temperature of extremal points and generalized off-shell Gibbs
free energy, respectively. Black holes are classified into different
topological classes based on their topological number. In this study, we found
unlike the charged AdS black hole, the charged GB AdS black hole exhibits a
critical point. Our findings reveal the occurrence of a liquid/gas-like
first-order phase transition between small-large black hole phases of the
spherical charged GB AdS black hole. We conclude that the charged GB AdS and
charged AdS black holes belong to different topological classes in the grand
canonical ensemble. Furthermore, connecting with the previous studies, we
conclude that the charged AdS and charged GB AdS black holes in canonical and
charged GB in the grand canonical ensemble belong to the same topological
classes.
