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.