  In this study, we consider a cosmological model for the Maxwell gravity which
is constructed by gauging the semi-simple extended Poincar\'e algebra. Inspired
by the Einstein-Yang-Mills theory, we describe the Maxwell gauge field in terms
of two additional time-dependent scalar fields. Within the context of a
homogeneous and isotropic Friedmann-Lema\^itre-Robertson-Walker universe, we
derive the Friedmann equations together with new contributions. Additionally,
we find the exact expressions for the additional scalar fields, considering the
exponential evolution of the scale factor. Moreover, we investigate the gauge
theory of gravity based on the Maxwell algebra and demonstrate that this model
leads to the (anti) de Sitter universe as well as a non-accelerated universe
model.
