  Entanglement entropy in the vacuum state of local field theories exhibits an
area law. However, nonlocal theories at large N and strong coupling violate
this area law. In these theories, the leading divergence in the entanglement
entropy is extensive for regions smaller than the effective nonlocality scale
and proportional to this effective nonlocality scale for regions larger than
it. This raises the question: is a volume law a generic feature of nonlocal
theories, or is it only present at strong coupling and large N?
  This paper investigates entanglement entropy of large regions in weakly
coupled nonlocal theories, to leading order in the coupling. The two theories
studied are phi^4 theory on the noncommutative plane and phi^4 theory with a
dipole type nonlocal modification using a fixed nonlocality scale. Both
theories are found to follow an area law to first order in the coupling, hence
no evidence is found for a volume law. This indicates that, perturbatively the
nonlocal interactions considered are not generating sufficient entanglement at
distances of the nonlocality scale to change the leading divergence, at least
to first order in the coupling. An argument against volume laws at higher
orders is also presented.
