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 in a one-dimensional framework. 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., 2025]. A second-order extension preserving the properties of the first-order scheme is given. The scheme is provably entropy-stable and positivity-preserving for all thermodynamic variables. Numerical test cases illustrate the performance and entropy stability of the new scheme, using six different equations of state as examples, four analytic and two tabulated ones.
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