  We investigate the entanglement entropy in gravity duals of confining large
$N_c$ gauge theories using the proposal of arXiv:hep-th/0603001,
arXiv:hep-th/0605073. Dividing one of the directions of space into a line
segment of length $l$ and its complement, the entanglement entropy between the
two subspaces is given by the classical action of the minimal bulk hypersurface
which approaches the endpoints of the line segment at the boundary. We find
that in confining backgrounds there are generally two such surfaces. One
consists of two disconnected components localized at the endpoints of the line
segment. The other contains a tube connecting the two components. The
disconnected surface dominates the entropy for $l$ above a certain critical
value $l_{\rm crit}$ while the connected one dominates below that value. The
change of behavior at $l=l_{\rm crit}$ is reminiscent of the finite temperature
deconfinement transition: for $l < l_{\rm crit}$ the entropy scales as $N_c^2$,
while for $l > l_{\rm crit}$ as $N_c^0$. We argue that a similar transition
should occur in any field theory with a Hagedorn spectrum of non-interacting
bound states. The requirement that the entanglement entropy has a phase
transition may be useful in constraining gravity duals of confining theories.
