  A modification of General Relativity that is based on the gravitational
Standard-Model Extension and incorporates nondynamical background fields has
recently been studied via the ADM formalism. Our objective in this paper is to
develop a better understanding of the additional contributions that arise on
the spacetime boundary $\partial M$. An extension of the previously introduced
boundary terms, which are relevant in the context of asymptotically flat
spacetimes, follows from the decomposition of $\partial M$ into timelike and
spacelike hypersurfaces. Furthermore, we present an alternative method of
deriving the field equations satisfied by the induced metric on the purely
spacelike hypersurfaces of the foliated spacetime. This leads to the dynamical
part of the Einstein equations modified by the background fields. Our results
have the potential to be applicable in various contexts such as modified black
holes and cosmology.
