  Phase transition of RN-AdS black hole is investigated from a new perspective.
Not only the cosmological constant is treated as pressure, but also the spatial
curvature of black hole is treated as topological charge. We obtain the
extended thermodynamic first law from which the mass is naturally viewed as
enthalpy instead of energy. In canonical ensemble with fixed topological charge
and electric charge, interesting van der Waals system's oscillatory behavior in
$T-S$, $P-V$ graphes and swallow tail behavior in $G-T$, $G-P$ graphes are
observed. By applying the Maxwell equal area law and by analysing the gibbs
free energy, we obtain analytical phase transition coexistence curves ($T-P$)
which are consistent with each other.
