In a wide class of cosmological models, a positive cosmological constant drives cosmic evolution toward an asymptotically de Sitter phase. Here we connect this behaviour to the increase of entropy over time, based on the idea that de Sitter is a maximum-entropy state. I will discuss how to prove a cosmic no-hair theorem for Robertson-Walker and Bianchi I spacetimes that admit a Q-screen with certain entropic properties: If generalized entropy increases up to a finite maximum value along the screen, then the spacetime is asymptotically de Sitter in the future. The theorem follows without using the Einstein field equations or assuming 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.