The grand potential

The grand potential is

$$\Phi_G\equiv E-TS-\mu N\qquad\hbox{so}\qquad {\rm d}\Phi_G=- S {\rm d}T-P {\rm d}V-N{\rm d}\mu$$

So $\Phi_G=\Phi_G(T,V,\mu)$. Note that two of the variables $T$ and $\mu$ are intensive, so $\Phi_G$, being extensive, must be simply proportional to $V$, the only extensive one: $\Phi_G=V\phi_G(T,\mu)$. But

$$\begin{eqnarray*} \phi_G(T,\mu)\!\!\!&=&\!\!\!\left({\partial \Phi_G\over\partia... ...\!\scriptstyle T,\mu}=-P\\ \Rightarrow\Phi_G\!\!\!&=&\!\!\!-PV. \end{eqnarray*}$$

This also follows from the fact that $\mu$ is just the Gibbs free energy per particle (see here), so $\mu N=G=E - TS + PV$ and hence $PV=-(E-TS-\mu N)=-\Phi_G$.

The fact that $\Phi_G$ is so simple doesn't lessen its formal utility in statistical mechanics.