  In this paper, we will consider cosmological implications of the Maxwell
theory coupled to a non-local $U(1)$-symmetric term. It is well-known that the
theory in flat space time, reduces to the Proca theory. However, it will be
shown that in curved space time the resulting theory will differ from the
coupled Einstein-Proca system. The cosmological perturbations on top of the de
Sitter space-time is also considered and the dynamics of separate modes will be
investigated in detail. Anisotropic cosmology of the model is also investigated
and we will show that the behavior of the universe at late time satisfies the
observational data and the model predicts an isotropic universe.
