  We study how entanglement spreads in the boundary duals of finite-cutoff
three-dimensional theories with positive, negative and zero cosmological
constant, the $T \bar{T} + \Lambda_{2}$ two-dimensional theories. We first
study the Hawking-Page transition in all three cases, and find that there is a
transition in all three scenarios at the temperature where the lengths of the
two cycles of the torus are the same. We then study the entanglement entropy in
the thermofield double states above the Hawking-Page transition, of regions
symmetrically placed on the two boundaries. We consider the case where the
region is one interval on each side, and the case where it is two intervals on
each side. We give an entanglement tsunami interpretation of the time-evolution
of the entanglement entropies.
