  Recent results suggest that new corrections to holographic entanglement
entropy should arise near phase transitions of the associated Ryu-Takayanagi
(RT) surface. We study such corrections by decomposing the bulk state into
fixed-area states and conjecturing that a certain `diagonal approximation' will
hold. In terms of the bulk Newton constant $G$, this yields a correction of
order $O(G^{-1/2})$ near such transitions, which is in particular larger than
generic corrections from the entanglement of bulk quantum fields. However, the
correction becomes exponentially suppressed away from the transition. The net
effect is to make the entanglement a smooth function of all parameters, turning
the RT `phase transition' into a crossover already at this level of analysis.
  We illustrate this effect with explicit calculations (again assuming our
diagonal approximation) for boundary regions given by a pair of disconnected
intervals on the boundary of the AdS$_3$ vacuum and for a single interval on
the boundary of the BTZ black hole. In a natural large-volume limit where our
diagonal approximation clearly holds, this second example verifies that our
results agree with general predictions made by Murthy and Srednicki in the
context of chaotic many-body systems. As a further check on our conjectured
diagonal approximation, we show that it also reproduces the $O(G^{-1/2})$
correction found Penington et al for an analogous quantum RT transition. Our
explicit computations also illustrate the cutoff-dependence of fluctuations in
RT-areas.
