  We study the late cosmological evolution, from the nonrelativistic matter
dominated era to the dark energy era, in modified gravity models described by
Degenerate Higher-Order Scalar-Tensor (DHOST) theories. They represent the most
general scalar-tensor theories propagating a single scalar degree of freedom
and include Horndeski and Beyond Horndeski theories. We provide the homogeneous
evolution equations for any quadratic DHOST theory, without restricting
ourselves to theories where the speed of gravitational waves coincides with
that of light since the present constraints apply to wavelengths much smaller
than cosmological scales. To illustrate the potential richness of the
cosmological background evolution in these theories, we consider a simple
family of shift-symmetric models, characterized by three parameters and compute
the evolution of dark energy and of its equation of state. We also identify the
regions in parameter space where the models are perturbatively stable.
