  We compute the entanglement entropy of a wide class of exactly solvable
models which may be characterized as describing matter coupled to gauge fields.
Our principle result is an entanglement sum rule which states that entropy of
the full system is the sum of the entropies of the two components. In the
context of the exactly solvable models we consider, this result applies to the
full entropy, but more generally it is a statement about the additivity of
universal terms in the entropy. We also prove that the Renyi entropy is exactly
additive and hence that the entanglement spectrum factorizes. Our proof
simultaneously extends and simplifies previous arguments, with extensions
including new models at zero temperature as well as the ability to treat finite
temperature crossovers. We emphasize that while the additivity is an exact
statement, each term in the sum may still be difficult to compute. Our results
apply to a wide variety of phases including Fermi liquids, spin liquids, and
some non-Fermi liquid metals.
