  For an affine toric variety $\mathrm{Spec}(A)$, we give a convex geometric
description of the Hodge decomposition of its Hochschild cohomology. Under
certain assumptions we compute the dimensions of the Hodge summands
$T^1_{(i)}(A)$, generalizing the existing results about the Andre-Quillen
cohomology group $T^1_{(1)}(A)$. We prove that every Poisson structure on a
possibly singular affine toric variety can be quantized in the sense of
deformation quantization.
