  Whether fast cooling processes occur or not is crucial for the thermal
evolution of neutron stars. In particular, the threshold of the direct Urca
process, which is one of the fast cooling processes, is determined by the
interior proton fraction $Y_p$, or the nuclear symmetry energy. Since recent
observations indicate the small radius of neutron stars, a low value is
preferred for the symmetry energy. In this study, simulations of neutron star
cooling are performed adopting three models for equation of state (EoS):
Togashi, Shen, and LS220 EoSs. The Togashi EoS has been recently constructed
with realistic nuclear potentials under finite temperature, and found to
account for the small radius of neutron stars. As a result, we find that, since
the direct Urca process is forbidden, the neutron star cooling is slow with use
of the Togashi EoS. This is because symmetry energy of Togashi EoS is lower
than those of other EoSs. Hence, in order to account for observed age and
surface temperature of isolated neutron stars (INS) with use of the Togashi
EoS, other fast cooling processes are needed regardless of the surface
composition.
