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

Warm asymmetric quark matter and proto-quark stars within the confined-isospin-density-dependent mass model