Let $X$ be an algebraic variety (or a complex manifold) over $\mathbb C$. Let $D(X)$ be its algebra of differential operators. Mostly I am interested in algebraic differential operators, but the case of smooth coefficients is also interesting(though the computation of Hochschild/cyclic (co)homology for these options coincide).

What is known about cohomology of $D(X)$ of this algebra viewed as a Lie algebra?

I know how to compute cyclic and Hohschild (co)homology of $D(X)$ and there is a theorem of Loday and Quillen which relates Lie algebra homology of $\mathfrak{gl}_{\infty}(A)$ to cyclic homology of $A$ for any algebra $A$, but Lie algebra homology of $A$ and $\mathfrak{gl}_{\infty}(A)$ may differ greately as some cycles may die on the infinite level.