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Protein condensates formed by liquid-liquid phase separation exhibit aging rheological properties and are considered under biological regulation and related to multiple diseases. In this work we provide a mesoscopic framework to describe the rheology of protein condensates. A phase transition from a nonaging phase to an aging phase occurs when the temperature is below a critical value. In the aging phase, we find that protein condensates behave as Maxwell fluids at all ages in the linear viscoelastic regime and the viscosities increase linearly over time. We also show that protein condensates are non-Newtonian fluids under a constant shear rate and the stress-strain curve exhibits a power-law scaling at large strain. Our theories are consistent with experimental observations and also make general predictions that are experimentally testable.