# Questions about thermodynamic potential and functional measure

In quantum field theory for condensed matter physics, an observable $$\hat{O}$$ may be expressed as: $$\langle\hat{O}\rangle=\frac{\int D(\psi,\psi^{\dagger}) \hat{O}(\psi,\psi^{\dagger})e^{-S(\psi,\psi^{\dagger})}}{\int D(\psi,\psi^{\dagger}) e^{-S(\psi,\psi^{\dagger})}}.\tag{1}$$

Because of formula of (1), an arbitary factor can be added to the functional measure, $$D(\psi,\psi^{\dagger})\rightarrow aD(\psi,\psi^{\dagger}),\tag{2}$$ without changing the physical results.

However, I read in most textbooks that thermodynamical potential $$\Omega=-\beta^{-1}\ln{\int D(\psi,\psi^{\dagger}) e^{-S(\psi,\psi^{\dagger})}}\tag{3}$$

Since thermodynamical potential is a function of $$T$$ (or $$\beta$$), (2) would result a shift of entropy: $$S\rightarrow S-\frac{\partial \frac{a}{T}}{\partial T}$$

So I think (3) is not complete and should be modified as some form of (1). Yet I couldn't write it down. Could anyone help?