  We extend standard results for vacuum asymptotically locally AdS (AlAdS)
spacetimes, showing that such spacetimes can be constructed as foliations where
the induced metric on each hypersurface satisfies Einstein's equation with
stress-energy. By an appropriate choice of stress-energy on the hypersurfaces,
the resulting AlAdS spacetime satisfies Einstein's equation with a negative
cosmological constant and physical stress tensor. We use this construction to
obtain AlAdS solutions whose boundaries are FRW cosmologies sourced by a
massless scalar field or by a perfect fluid obeying the strong energy
condition. We focus on FRW universes that approach Minkowski spacetime at late
times, yielding AlAdS spacetimes that approach either the Poincar\'e patch of
pure AdS or the AdS soliton, which we view as late time equilibrium states. As
an application of these solutions, we use the AdS/CFT correspondence to study
the approach to equilibrium of the entanglement entropy and of the boundary
stress tensor of the boundary CFT. We find that the energy of the
asymptotically AdS solitonic solution is consistent with the conjecture that
the AdS soliton is the lowest-energy solution to Einstein's equation with
negative cosmological constant. The time dependent correction to the
entanglement entropy is found to decay like a power law, with rate set by the
Hubble parameter and the power determined by the equation of state of the
cosmic fluid.
