  We develop a variational approximation to the entanglement entropy for scalar
$\phi^4$ theory in 1+1, 2+1, and 3+1 dimensions, and then examine the
entanglement entropy as a function of the coupling. We find that in 1+1 and 2+1
dimensions, the entanglement entropy of $\phi^4$ theory as a function of
coupling is monotonically decreasing and convex. While $\phi^4$ theory with
positive bare coupling in 3+1 dimensions is thought to lead to a trivial free
theory, we analyze a version of $\phi^4$ with infinitesimal negative bare
coupling, an asymptotically free theory known as precarious $\phi^4$ theory,
and explore the monotonicity and convexity of its entanglement entropy as a
function of coupling. Within the variational approximation, the stability of
precarious $\phi^4$ theory is related to the sign of the first and second
derivatives of the entanglement entropy with respect to the coupling.
