  The possibility of a neutron m=2-superfluid in the interior of neutron stars
is investigated. This pairing state is energetically favoured in strong
magnetic fields ($H\sim 10^{16}-10^{17}$ G). Because of the node in the
angular-dependent energy gap along the field direction the neutrino emissivity
is only suppressed polynomially as function in $T/T_{c}$ instead of
exponentially, as it is obtained for a nodeless pairing state. The effect of
this pairing state on the thermal evolution of neutron stars is studied, and
its outcome is compared with the evolution of ``normal'', i.e. nodeless,
superfluid and non-superfluid neutron stars, and also with observations. We
find that particularly the predicted surface temperatures of the enhanced
cooling scenario considerably change and come into agreement with temperatures
deduced from observational data within the hydrogen atmosphere model.
Furthermore the surface temperature depends on the magnetic field strength as
an additional parameter aside from the neutron star mass. The latter is however
only operative in the case of the intermediate cooling scenario.
