  We compute the topological entanglement entropy for a large set of lattice
models in $d$-dimensions. It is well known that many such quantum systems can
be constructed out of lattice gauge models. For dimensionality higher than two,
there are generalizations going beyond gauge theories, which are called higher
gauge theories and rely on higher-order generalizations of groups. Our main
concern is a large class of $d$-dimensional quantum systems derived from
Abelian higher gauge theories. In this paper, we derive a general formula for
the bipartition entanglement entropy for this class of models, and from it we
extract both the area law and the sub-leading terms, which explicitly depend on
the topology of the entangling surface. We show that the entanglement entropy
$S_A$ in a sub-region $A$ is proportional to $\log(GSD_{\tilde{A}})$, where
\(GSD_{\tilde{A}}\) is the ground state degeneracy of a particular restriction
of the full model to \(A\). The quantity $GSD_{\tilde{A}}$ can be further
divided into a contribution that scales with the size of the boundary $\partial
A$ and a term which depends on the topology of $\partial A$. There is also a
topological contribution coming from $A$ itself, that may be non-zero when $A$
has a non-trivial homology. We present some examples and discuss how the
topology of $A$ affects the topological entropy. Our formalism allows us to do
most of the calculation for arbitrary dimension $d$. The result is in agreement
with entanglement calculations for known topological models.
