  The thermal conductivity of the color-flavor locked phase of dense quark
matter is calculated. The dominant contribution to the conductivity comes from
photons and Nambu-Goldstone bosons associated with the breaking of baryon
number, both of which are trapped in the quark core. Because of their very
large mean free path the conductivity is also very large. The cooling of the
quark core arises mostly from the heat flux across the surface of direct
contact with the nuclear matter. As the thermal conductivity of the neighboring
layer is also high, the whole interior of the star should be nearly isothermal.
Our results imply that the cooling time of compact stars with color-flavor
locked quark cores is similar to that of ordinary neutron stars.
