  We give a cohomological and geometrical interpretation for the weighted
Ehrhart theory of a full-dimensional lattice polytope $P$, with Laurent
polynomial weights of geometric origin. For this purpose, we calculate the
motivic Chern and Hirzebruch characteristic classes of a mixed Hodge module
complex $\mathcal{M}$ whose underlying cohomology sheaves are constant on the
$\mathbb{T}$-orbits of the toric variety $X_P$ associated to $P$. Besides
motivic coefficients, this also applies to the intersection cohomology Hodge
module. We introduce a corresponding generalized Hodge $\chi_y$-polynomial of
the ample divisor $D_P$ on $X_P$. Motivic properties of these characteristic
classes are used to express this Hodge polynomial in terms of a very general
weighed lattice point counting and the corresponding weighted Ehrhart theory.
We introduce, for such a mixed Hodge modules complex $\mathcal{M}$ on $X$, an
Ehrhart polynomial $E_{P,\mathcal{M}}$ generalizing the Hodge polynomial of
$\mathcal{M}$ and satisfying a reciprocity formula and a purity formula fitting
with the duality for mixed Hodge modules. This Ehrhart polynomial and its
properties depend only on a Laurent polynomial weight function on the faces $Q$
of $P$. In the special case of the intersection cohomology mixed Hodge module,
the weight function corresponds to Stanley's $g$-function of the polar polytope
of $P$, hence it depends only on the combinatorics of $P$. In particular, we
obtain a combinatorial formula for the intersection cohomology signature.
