  We perform a phase space analysis of a generalized modified gravity theory
with nonminimally coupling between geometry and matter. We apply the dynamical
system approach to this generalized model and find that in the cosmological
context, different choices of Lagrangian density will apparently result in
different phases of the Universe. By carefully choosing the variables, we prove
that there is an attractor solution to describe the late time accelerating
universe when the modified gravity is chosen in a simple power-law form of the
curvature scalar. We further examine the temperature evolution based on the
thermodynamic understanding of the model. Confronting the model with supernova
type Ia data sets, we find that the nonminimally coupled theory of gravity is a
viable model to describe the late time Universe acceleration.
