  We show that within a recently developed nonlocal chiral quark model the
critical density for a phase transition to color superconducting quark matter
under neutron star conditions can be low enough for these phases to occur in
compact star configurations with masses below 1.3 M_sun. We study the cooling
of these objects in isolation for different values of the gravitational mass
and argue that, if the quark matter phase would allow unpaired quarks, the
corresponding hybrid stars would cool too fast. The comparison with
observational data puts tight constraints on possible color superconducting
quark matter phases. Possible candidates with diquark gaps of the order of 10
keV - 1 MeV such as the "2SC+X" and the color spin locking (CSL) phase are
presented.
