  The topological charge $\epsilon$ of AdS black hole is introduced in
Ref.[1,2], where a complete thermodynamic first law is obtained. In this paper,
we investigate a new phase transition related to the topological charge in
Einstein-Maxwell theory. Firstly, we derive the explicit solutions
corresponding to the divergence of specific heat $C_{\epsilon}$ and determine
the phase transition critical point. Secondly, the $T-r$ curve and $T-S$ curve
are investigated and they exhibit an interesting van der Waals system's
behavior. Critical physical quantities are also obtained which are consistent
with those derived from the specific heat analysis. Thirdly, a van der Waals
system's swallow tail behavior is observed when $\epsilon>\epsilon_{c}$ in the
$F-T$ graph. What's more, the analytic phase transition coexistence lines are
obtained by using the Maxwell equal area law and free energy analysis, the
results of which are consistent with each other.
