Evolution of the internal temperature T and the chemical imbalances η_npe and η_npμ with the initial condition T = 10⁸ K and null chemical imbalances. The spin-down is assumed to be caused by the magnetic dipole radiation, with dipolar field strength B = 2.8 × 10⁸ G and initial period of P₀ = 1 ms. The dashed lines indicate the energy gap Δ_n = 0.05 MeV. Upper panel: evolution of a 1.69 M_⊙ star built with the BPAL21 EOS (Prakash et al. 1988), which allows muon direct Urca reactions, and we add the vortex creep reheating term of Eq. (27) with an excess of angular momentum J = 10⁴³ erg s. Lower panel: evolution of a 1.68 M_⊙ star built with the BPAL21 EOS (Prakash et al. 1988), which forbids direct Urca reactions with muons.