  We consider cooling of neutron stars (NSs) with superfluid cores composed of
neutrons, protons, and electrons (assuming singlet-state pairing of protons,
and triplet-state pairing of neutrons). We mainly focus on (nonstandard)
triplet-state pairing of neutrons with the $|m_J| = 2$ projection of the total
angular momentum of Cooper pairs onto quantization axis. The specific feature
of this pairing is that it leads to a power-law (nonexponential) reduction of
the emissivity of the main neutrino processes by neutron superfluidity. For a
wide range of neutron critical temperatures $T_{cn}$, the cooling of NSs with
the $|m_J| = 2$ superfluidity is either the same as the cooling with the $m_J =
0$ superfluidity, considered in the majority of papers, or much faster. The
cooling of NSs with density dependent critical temperatures $T_{cn}(\rho)$ and
$T_{cp}(\rho)$ can be imitated by the cooling of the NSs with some effective
critical temperatures $T_{cn}$ and $T_{cp}$ constant over NS cores. The
hypothesis of strong neutron superfluidity with $|m_J| = 2$ is inconsistent
with current observations of thermal emission from NSs, but the hypothesis of
weak neutron superfluidity of any type does not contradict to observations.
