Rotational enhancement and stability of protoquark stars during thermal evolution

We present the first systematic study of rigidly rotating protoquark stars based on isentropic equations of state (EOS) within the density-dependent quark mass (DDQM) framework. Using a quasi-static equilibrium approach, we follow the Kelvin--Helmholtz evolution from hot, lepton-rich matter to a cold, catalyzed quark star (QS). Rotation substantially enhances the maximum stable mass (by up to $\sim 40\%$), equatorial radius, and key rotational observables, with the ratio of rotational kinetic to gravitational potential energy, $T_{\rm kin}/|W|$, reaching $0.18$--$0.19$ near the Keplerian limit, indicating a heightened susceptibility to gravitational-wave--emitting instabilities. Thermal evolution introduces a clear ordering: all stellar properties peak during the lepton-rich stages and decrease monotonically as the star cools. Compared to hadronic stars, rotating proto-QSs exhibit larger radii, higher moments of inertia, and stronger quadrupolar deformation, producing a distinct signature in the mass--radius--spin plane. The EOS parameters are constrained using current astrophysical observations, including mass--radius measurements from HESS~J1731--347 and PSR~J0030+0451, the high-mass constraint from PSR~J0740+6620, and mass-radius constraints inferred from GW170817. The results demonstrate that future multimessenger observations must account for both thermal history and rotation to identify quark matter (QM) in compact stars robustly.