In this study, we consider constructing a cosmological model for the Maxwell
gravity. Inspired by the Einstein-Yang-Mills theory, we describe the Maxwell
gauge field in terms of two time-dependent scalar fields. For a cosmological
scenario, we use a homogeneous and isotropic
Friedmann-Lemaitre-Robertson-Walker universe and derive the Friedmann equations
together with new contributions. Then we show that the minimal model of the
Maxwell gravity describes an expanding universe without acceleration together
with a constant equation of state parameter in a special condition.