  We compute the thermal conductivity of color-flavor locked (CFL) quark
matter. At temperatures below the scale set by the gap in the quark spectrum,
transport properties are determined by collective modes. In this work we focus
on the contribution from the lightest modes, the superfluid phonon and the
massive neutral kaon. The calculation is done in the framework of kinetic
theory, using variational solutions of the linearized Boltzmann equation. We
find that the thermal conductivity due to phonons is \kappa^P =1.04 10^{26}
mu_{500}^8 \Delta_{50}^{-6} erg/(cm s K), where \mu_{500} is the chemical
potential in units of 500 MeV and \Delta_{50} is the gap in units of 50 MeV.
The contribution of kaons is \kappa^K = 2.81 10^{21} f_{\pi,100}^4
T_{MeV}^{1/2} m_{10}^{-5/2} erg/(cm s K), where f_{\pi,100} is the pion decay
constant in units of 100 MeV, T_{MeV} is the temperature in units of 1 MeV, and
m_{10} is the kaon mass in units of 10 MeV. These values are smaller than
previous estimates, but still much larger than (in the case of phonons) or
similar to (for kaons) the corresponding values in nuclear matter. From the
phonon thermal conductivity we estimate that a CFL quark matter core of a
compact star becomes isothermal on a timescale of a few seconds.
