  We consider deformations of a pair $(X,\partial X)$, where $X$ is an affine
toric Gorenstein variety and $\partial X$ is its boundary. We compute the
tangent and obstruction space for the corresponding deformation functor and for
an admissible lattice degree $m$ we construct the miniversal deformation of
$(X,\partial X)$ in degrees $-km$, for all $k\in \mathbb{N}$ together. This in
particular generalizes Altmann's construction of the miniversal deformation of
an isolated Gorenstein toric singularity to an arbitrary non-isolated
Gorenstein toric singularity. We see that it is more natural to work with
deformations of a pair $(X,\partial X)$ rather than only with deformations of
$X$, since many computations are in fact easier.
