  We initiate a systematic study of the cohomology of cluster varieties. We
introduce the Louise property for cluster algebras that holds for all acyclic
cluster algebras, and for most cluster algebras arising from marked surfaces.
For cluster varieties satisfying the Louise property and of full rank, we show
that the cohomology satisfies the curious Lefschetz property of Hausel and
Rodriguez-Villegas, and that the mixed Hodge structure is split over the
rationals. We give a complete description of the highest weight part of the
mixed Hodge structure of these cluster varieties, and develop the notion of a
standard differential form on a cluster variety. We show that the point counts
of these cluster varieties over finite fields can be expressed in terms of
Dirichlet characters. Under an additional integrality hypothesis, the point
counts are shown to be polynomials in the order of the finite field.
