  In normal degenerate quark matter, the exchange of dynamically screened
transverse gluons introduces infrared divergences in the quark self-energies
that lead to the breakdown of the Fermi liquid description. If the core of
neutron stars are composed of quark matter with a normal component, cooling by
direct quark Urca processes may be modified by non-Fermi liquid corrections. We
find that while the quasiparticle density of states is finite and non-zero at
the Fermi surface, its frequency derivative diverges and results in non-Fermi
liquid corrections to the specific heat of the normal, degenerate component of
quark matter. We study these non-perturbative non-Fermi liquid corrections to
the specific heat and the temperature dependence of the chemical potential and
show that these lead to a reduction of the specific heat.
