  We model the quiescent luminosity of accreting neutron stars with several
equation of states (EOSs), including the effect of pion condensation and
superfluidity. As a consequence of comparison with the observations, we show
that the results with Togashi EoS (the strong direct Urca process is forbidden)
and TM1e EoS (mass at direct Urca process is $2.06 M_\odot$) can explain the
observations well by considering pion condensation and the effect of
superfluidity, while LS220 EoS and TM1 EoS can explain the observations well by
considering the baryon direct Urca process and the effect of superfluidity.
Besides, we compare the results with the observations of a neutron star RX
J0812.4-3114 which has the low average mass accretion rate
($\langle\dot{M}\rangle\sim(4-15)\times 10^{-12}~M_\odot ~\rm yr^{-1}$) but
high thermal luminosity ($L_q^\infty\sim(0.6-3)\times10^{33}~\rm erg ~
s^{-1}$), and we suggest that a low-mass neutron star ($<1M_\odot$) with
minimum cooling can explain the lower limit of the observation of thermal
luminosity of RX J0812.4-3114, which is qualitatively consistent with the
previous work~\cite{Zhao2019}. However, to explain its upper limit, some other
heating mechanisms besides standard deep crustal heating may be needed.
