The minimal cooling paradigm for neutron star cooling assumes that enhanced cooling due to neutrino emission from any direct Urca process, due either to nucleons or to exotica such as hyperons, Bose condensates, or deconfined quarks, does not occur. This scenario was developed to replace and extend the so-called standard cooling scenario to include neutrino emission from the Cooper pair breaking and formation processes that occur near the critical temperature for superfluid/superconductor pairing. Recently, it has been found that Cooper-pair neutrino emission from the vector channel is suppressed by a large factor compared to the original estimates that violated vector current conservation. We show that Cooper-pair neutrino emission remains, nevertheless, an efficient cooling mechanism through the axial channel. As a result, the elimination of neutrino emission from Cooper-paired nucleons through the vector channel has only minor effects on the long-term cooling of neutron stars within the minimal cooling paradigm. We further quantify precisely the effect of the size of the neutron 3P2 gap and demonstrate that consistency between observations and the minimal cooling paradigm requires that the critical temperature T_c for this gap covers a range of values between T_c^min < 0.2 x 10^9 K up to T_c^max > 0.5 times 10^9 K in the core of the star. In addition, it is required that young neutron stars have heterogenous envelope compositions: some must have light-element compositions and others must have heavy-element compositions. Unless these two conditions are fulfilled, about half of the observed young cooling neutron stars are inconsistent with the minimal cooling paradigm and provide evidence for the existence of enhanced cooling.
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