  The effect of color superconductivity on the cooling of quark stars and
neutron stars with large quark cores is investigated. Various known and new
quark-neutrino processes are studied. As a result, stars being in the color
flavor locked (CFL) color superconducting phase cool down extremely fast. Quark
stars with no crust cool down too rapidly in disagreement with X-ray data. The
cooling of stars being in the N_f =2 color superconducting (2SC) phase with a
crust is compatible with existing X-ray data. Also the cooling history of stars
with hypothetic pion condensate nuclei and a crust does not contradict the
data.
