  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.
