Topology of 5-dimensional Einstein–Gauss–Bonnet AdS black hole thermodynamics surrounded by a cloud of Strings

We present a topological analysis of five-dimensional Einstein–Gauss–Bonnet (EGB) AdS black holes surrounded by a cloud of strings, employing Duan’s topological current $\varphi$-mapping theory to classify thermodynamic critical points in a parameter-independent manner. We used two different approaches respectively considering Duan’s potential and the generalized potential. Our results demonstrate that the sign of the Gauss–Bonnet coupling constant $\alpha$ fundamentally determines the topology of the thermodynamic state space: $\alpha >0$ yields a single critical point with topological charge $+1$, while $\alpha <0$ produces two critical points with opposite charges $(+1,-1)$, giving a total charge of zero. The string cloud parameter $a$ influences the positions of these critical points but leaves their topological charges unchanged, clearly distinguishing the roles of higher-curvature corrections and surrounding matter fields. Moreover, on considering the generalized potential, results shows topological number $W=+1$ for $\alpha >0$ and $W=0\text{or}+1$ for $\alpha <0$ indicating the clear influence of the Gauss–Bonnet coupling constant $a$ in the topological thermodynamic classification. This framework offers a universal classification scheme for black hole thermodynamics and provides deeper insight into how modified gravity and external matter distributions shape the underlying topological structure.