Is it common to use algebraic geometry for statistical mechanics?
More specifically, can we consider the phase space hypersurface consistent with a level of energy as a variety in the affine space and study its properties including volume used to define entropy, by using the mapping to the ideals and radical ideals structure in the ring of polynomials?
Spheres with different radii have different volumes but are isomorphic as varieties (and as schemes), so no, you cannot define volume in terms of purely algebro-geometric invariants.
I don't know about about varieties in statistical mechanics. However, moduli spaces in physics need not be smooth and they can be studied by a generalisation of varieties to the $C^{\infty}$ setting. These are spaces cut out by the locus of a collection of smooth functions over some point (usually zero) to $\mathbb{R}$. They model manifolds with corners and more exotic singularities. The category of such have good properties, for example fibre products without a transversality condition. See the papers by Joyce and Francis-Staite for the maths.
