  We analyze the phase structure of $d=4$ $\mathcal{N}=2$ large $N$ SYM theory
with flavor on $S^1\times S^3$ by using geometric entropy as an order
parameter. We introduce chemical potential conjugate to global U(1) symmetry
and find the third order phase transition at finite density by using the
geometric entropy as the order parameter. We also find that the geometric
entropy at the finite density has an interesting behavior at low temperature
and for large $N_f$.
