  We extend the confined-isospin-density-dependent mass (CIDDM) model to
include temperature dependence of the equivalent mass for quarks. Within the
CIDDM model, we study the equation of state (EOS) for $\beta$-equilibrium quark
matter, quark symmetry energy, quark symmetry free energy, and the properties
of quark stars at finite temperature. We find that including the temperature
dependence of the equivalent mass can significantly influence the properties of
the strange quark matter (SQM) as well as the quark symmetry energy, the quark
symmetry free energy, and the maximum mass of quark stars at finite
temperature. The mass-radius relations for different stages of the proto-quark
stars (PQSs) along the star evolution are analyzed. Our results indicate that
the heating (cooling) process for PQSs will increase (decrease) the maximum
mass within CIDDM model by including temperature dependence of the equivalent
mass for quarks.
