  In this paper we study various aspects of entanglement entropy in
strongly-coupled de Sitter quantum field theories in various dimensions. We
find gravity solutions that are dual to field theories in a fixed de Sitter
background, both in equilibrium and out-of-equilibrium configurations. The
latter corresponds to the Vaidya generalization of the AdS black hole solutions
with hyperbolic topology. We compute analytically the entanglement entropy of
spherical regions and show that there is a transition when the sphere is as big
as the horizon. We also explore thermalization in time-dependent situations in
which the system evolves from a non-equilibrium state to the Bunch-Davies
state. We find that the saturation time is equal to the light-crossing time of
the sphere. This behavior is faster than random walk and suggests the existence
of free light-like degrees of freedom.
