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We study the structures of hybrid stars with leptons at finite temperatureunder beta equilibrium. For the quark phase, we use the three flavorNambu-Jona-Lasinio (NJL) model. For the hadron phase, we adopt nuclear equationof state (EOS) by Shen et al.. This EOS is in the framework of the relativisticmean field theory including the tree body effects. For the hadron-quark phasetransition, we impose the bulk Gibbs construction or the Maxwell constructionto take into account uncertainties by {\it finite size effects}. We find thatthe pure quark phase does not appear in stable star cores in all cases. Withthe phase transition, the maximum masses increase $\sim 10 %$ for high leptonfraction. On the contrary, without the transition, they decrease $\sim 10 %$.We also find that, in the NJL model, the lepton fraction is more important forstructures of unstable stars than the temperature. This result is important formany astrophysical phenomena such as the core collapse of massive stars.

Lepton effects on the proto-neutron stars with the hadron-quark mixed phase in the Nambu-Jona-Lasinio model