  Superfluidity of $\Lambda$ and $\Sigma^-$ admixed in neutron star (NS) cores
is investigated realistically for hyperon ($Y$)-mixed NS models obtained using
a $G$-matrix-based effective interaction approach. Numerical results for the
equation of state (EOS) with the mixing ratios of the respective components and
the hyperon energy gaps including the temperature dependence are presented.
These are meant to serve as physical inputs for $Y$-cooling calculations of
NSs. By paying attention to the uncertainties of the EOS and the $YY$
interactions, it is shown that both $\Lambda$ and $\Sigma^-$ are superfluid as
soon as they appear although the magnitude of the critical temperature and the
density region where superfluidity exists depend considerably on the $YY$
pairing potential. Considering momentum triangle condition and the occurrence
of superfluidity, it is found that a so-called `` hyperon
cooling\rq\rq~(neutrino-emission from direct Urca process including $Y$)
combined with $Y$-superfluidity may be able to account for observations of the
colder class of NSs. It is remarked that $\Lambda$-hyperons play a decisive
role in the hyperon cooling scenario. Some comments are given regarding the
consequences of the less attractive $\Lambda\Lambda$ interaction recently
suggested by the `` NAGARA event\rq\rq~$^6_{\Lambda\Lambda}$He.
