  Recent observational data on transiently-accreting neutron stars has
unequivocally shown fast-cooling sources, such as in the case of neutron star
MXB 1659-29. Previous calculations have estimated its total neutrino luminosity
and heat capacity, as well as suggested that direct Urca reactions take place
in $1 \%$ of the volume of the core. In this paper, we reproduce the inferred
luminosity of this source with detailed models of equations of state (EOS) and
nuclear pairing gaps. We show that three superfluidity gap models are
inconsistent with data for all EOS and another three are disfavoured because of
fine tuning arguments. We also calculate the total heat capacity for all
constructed stars and show that independent observations of mass and luminosity
could set constraints on the core superfluidity of a source as well as the
density slope of the symmetry energy, L. This is an important step towards
defining a universal equation of state for neutron stars and therefore, towards
a better understanding of the phase diagram of asymmetric matter at high
densities.
