  We study the thermal evolution of neutron stars containing deconfined quark
matter in their core. Such objects are generally referred to as quark-hybrid
stars. The confined hadronic matter in their core is described in the framework
of non-linear relativistic nuclear field theory. For the quark phase we use a
non-local extension of the SU(3) Nambu Jona-Lasinio model with vector
interactions. The Gibbs condition is used to model phase equilibrium between
confined hadronic matter and deconfined quark matter. Our study indicates that
high-mass neutron stars may contain between 35 and 40 % deconfined quark-hybrid
matter in their cores. Neutron stars with canonical masses of around $1.4\,
M_\odot$ would not contain deconfined quark matter. The central proton
fractions of the stars are found to be high, enabling them to cool rapidly.
Very good agreement with the temperature evolution established for the neutron
star in Cassiopeia A (Cas A) is obtained for one of our models (based on the
popular NL3 nuclear parametrization), if the protons in the core of our stellar
models are strongly paired, the repulsion among the quarks is mildly repulsive,
and the mass of Cas A has a canonical value of $1.4\, M_\odot$.
