  We study thermodynamic and transport properties for the isotropic
color-spin-locking (iso-CSL) phase of two-flavor superconducting quark matter
under compact star constraints within a NJL-type chiral quark model. Chiral
symmetry breaking and the phase transition to superconducting quark matter
leads to a density dependent change of quark masses, chemical potentials and
diquark gap. A self-consistent treatment of these physical quantities
influences on the microscopic calculations of transport properties. We present
results for the iso-CSL direct URCA emissivities and bulk viscosities, which
fulfill the constraints on quark matter derived from cooling and rotational
evolution of compact stars. We compare our results with the phenomenologically
successful, but yet heuristic 2SC+X phase. We show that the microscopically
founded iso-CSL phase can replace the purely phenomenological 2SC+X phase in
modern simulations of the cooling evolution for compact stars with color
superconducting quark matter interior.
