  For a conformal field theory (CFT) deformed by a relevant operator, the
entanglement entropy of a ball-shaped region may be computed as a perturbative
expansion in the coupling. A similar perturbative expansion exists for excited
states near the vacuum. Using these expansions, this work investigates the
behavior of excited state entanglement entropies of small, ball-shaped regions.
The motivation for these calculations is Jacobson's recent work on the
equivalence of the Einstein equation and the hypothesis of maximal vacuum
entropy [arXiv:1505.04753], which relies on a conjecture stating that the
behavior of these entropies is sufficiently similar to a CFT. In addition to
the expected type of terms which scale with the ball radius as $R^d$, the
entanglement entropy calculation gives rise to terms scaling as $R^{2\Delta}$,
where $\Delta$ is the dimension of the deforming operator. When
$\Delta\leq\frac{d}{2}$, the latter terms dominate the former, and suggest that
a modification to the conjecture is needed.
