  In a wide class of cosmological models, a positive cosmological constant
drives cosmological evolution toward an asymptotically de Sitter phase. Here we
connect this behavior to the increase of entropy over time, based on the idea
that de Sitter spacetime is a maximum-entropy state. We prove a cosmic no-hair
theorem for Robertson-Walker and Bianchi I spacetimes that admit a Q-screen
("quantum" holographic screen) with certain entropic properties: If generalized
entropy, in the sense of the cosmological version of the Generalized Second Law
conjectured by Bousso and Engelhardt, increases up to a finite maximum value
along the screen, then the spacetime is asymptotically de Sitter in the future.
Moreover, the limiting value of generalized entropy coincides with the de
Sitter horizon entropy. We do not use the Einstein field equations in our
proof, nor do we assume the existence of a positive cosmological constant. As
such, asymptotic relaxation to a de Sitter phase can, in a precise sense, be
thought of as cosmological equilibration.
