  For a smooth toric variety X over a field of positive characteristic, a
T-equivariant \'{e}tale cover Y \rightarrow T^*X^{(1)} trivializing the sheaf
of crystalline differential operators on X is constructed. This trivialization
is used to show that the sheaf of differential operators is a trivial Azumaya
algebra along the fibers of the moment map. This result is then extended to
certain Azumaya algebras on hypertoric varieties, whose global sections are
central reductions of the hypertoric enveloping algebra in positive
characteristic. A criteria for a derived Beilinson-Bernstein localization
theorem is then formulated.
