  We show that observations of the core temperature of transiently-accreting
neutron stars combined with observations of an accretion outburst give a lower
limit to the neutron star core heat capacity. For the neutron stars in the low
mass X-ray binaries KS 1731-260, MXB 1659-29, and XTE J1701-462, we show that
the lower limit is a factor of a few below the core heat capacity expected if
neutrons and protons in the core are paired, so that electrons provide the
dominant contribution to the heat capacity. This limit rules out a core
dominated by a quark color-flavor-locked (CFL) phase, which would have a much
lower heat capacity. Future observations of or limits on cooling during
quiescence will further constrain the core heat capacity.
