We present a new model-independent (applicable for a broad range of equations of state) analysis of the neutrino emissivity due to triplet neutron pairing in neutron star cores. We find that the integrated neutrino luminosity of the Cooper Pair Formation (CPF) process can be written as a product of two factors. The first factor depends on the neutron star mass, radius and maximal critical temperature of neutron pairing in the core, $T_{Cn mathrm{max}}$, but not on the particular superfluidity model; it can be expressed by an analytical formula valid for many nucleon equations of state. The second factor depends on the shape of the critical temperature profile within the star, the ratio of the temperature $T$ to $T_{Cn mathrm{max}}$, but not on the maximal critical temperature itself. While this second factor depends on the superfluidity model, it obeys several model-independent constraints. This property allows one to analyse the thermal evolution of neutron stars with superfluid cores without relying on a specific model of their interiors. The constructed expressions allow us to perform a self-consistent analysis of spectral data and neutron star cooling theory. We apply these findings to the cooling neutron star in the Cassiopeia A supernova remnant using 14 sets of observations taken over 19 years. We constrain $T_{Cnmathrm{max}}$ to the range of $ (5-10)times 10^8$ K. This value depends weakly on the equation of state and superfluidity model, and will not change much if cooling is slower than the current data suggest. We also constrain the overall efficiency of the CPF neutrino luminosity.
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