Hodge theory in general

Question

I know a bit of Hodge theory, and I know that there is an analogue in the symplectic case, where instead of inducing the $\star$-product using the metric we use the symplectic form. Is in true in general that if we have a linear isomorphism $$\phi:T^*M\longrightarrow TM,$$ then we can define an associated $\star$-product and derive a Hodge theory for it? In particular, I'm interested in the study of deRham cohomology using such a generalised theory.