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We calculate neutrino emissivities from the decay and scattering of Goldstone bosons in the color-flavor-locked (CFL) phase of quarks at high baryon density. Interactions in the CFL phase are described by an effective low-energy theory. For temperatures in the tens of keV range, relevant to the long-term cooling of neutron stars, the emissivities involving Goldstone bosons dominate over those involving quarks, because gaps in the CFL phase are $\sim 100$ MeV while the masses of Goldstone modes are on the order of 10 MeV. For the same reason, the specific heat of the CFL phase is also dominated by the Goldstone modes. Notwithstanding this, both the emissivity and the specific heat from the massive modes remain rather small, because of their extremely small number densities. The values of the emissivity and the specific heat imply that the timescale for the cooling of the CFL core in isolation is $\sim 10^{26}$ y, which makes the CFL phase invisible as the exterior layers of normal matter surrounding the core will continue to cool through significantly more rapid processes. If the CFL phase appears during the evolution of a proto-neutron star, neutrino interactions with Goldstone bosons are expected to be significantly more important since temperatures are high enough ($\sim 20-40$ MeV) to admit large number densities of Goldstone modes.