  This paper is dedicated to scrutinizing the cosmology in massive gravity. A
matter field of the dark sector is coupled to an effective composite metric
while a standard matter field couples to the dynamical metric in the usual way.
For this purpose, we study the dynamical system of cosmological solutions by
using phase analysis, which provides an overview of the class of cosmological
solutions in this setup. This also permits us to study the critical points of
the cosmological equations together with their stability. We show the presence
of stable attractor de Sitter critical points relevant to the late-time cosmic
acceleration. Furthermore, we study the tensor, vector and scalar perturbations
in the presence of standard matter fields and obtain the conditions for the
absence of ghost and gradient instabilities. Hence, massive gravity in the
presence of the effective composite metric can accommodate interesting dark
energy phenomenology, that can be observationally distinguished from the
standard model according to the expansion history and cosmic growth.
