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

Thermal evolution of hybrid stars within the framework of a nonlocal Nambu--Jona-Lasinio model