The present work concerns the derivation of a fully well-balanced Godunov-type finite volume scheme for the Euler equations with a gravitational potential based on an approximate Riemann solver. It is an extension to general equations of states of the entropy-stable and fully well-balanced scheme for ideal gases recently forwarded in [Berthon et al., 2024]. When the system is equipped with a convex entropy and associated entropy inequality, it is also entropy-stable and positivity-preserving for all thermodynamic variables. An extension to high order accuracy is presented. Numerical test cases illustrate the performance of the new scheme, using six different equations of state as examples, four analytic and two tabulated ones.