  The presence of superfluidity in neutron star interiors can affect the
cooling of neutron stars in intricate ways, enhancing certain mechanisms and
suppressing others. Model calculations employing realistic nuclear potentials
in Bardeen-Cooper-Schrieffer theory generally suggest the development of a
$^3$P$_2$-$^3$F$_2$ pairing gap, and therefore the presence of superfluidity in
dense neutron-star matter.Improved models that go beyond conventional
mean-field calculations by including polarization effects suggest a suppression
of the triplet gap. We have evaluated the pairing interaction by summing the
``parquet'' Feynman diagrams which include both ladder and ring diagrams
systematically, {\em plus\} a set of important non-parquet diagrams, making
this the most comprehensive diagram-based approach presently available. Our
results suggest a radical suppression of the $^3$P$_2$-$^3$F$_2$ triplet
pairing gap and an enhancement of $^3$P$_0$ pairing.
