Topological Interpretation of Black Hole Phase Transition in Gauss-Bonnet Gravity

Phase transitions of Einstein-Gauss-Bonnet black holes are studied using Duan's $\phi-$ field topological current theory, where black holes are treated as topological defects in the thermodynamic parameter space. The kinetics of thermodynamic defects are studied using Duan's bifurcation theory. In this picture, a first-order phase transition between small/large black hole phases is interpreted as the interchange of winding numbers between the defects as a result of some action at a distance.We observe a first-order phase transition between small/large black holes for $D=5$ Gauss-Bonnet theory similar to Reissner-Nordstr\"{o}m black holes in AdS space. This implies that these black hole solutions share the same topology and phase structure. We have also studied the phase transition of neutral black holes in $D\geq 6$ and found a transition between unstable small and large stable black hole phases similar to the case of neutral black holes in AdS space. Recently, it has been conjectured that black holes with similar topological structure exhibit the same thermodynamic properties. Our results strengthen the conjecture by connecting the topological nature of black holes to phase transitions.