  The cooling of neutron stars by URCA processes in the kaon-condensed neutron
star matter for various forms of nuclear symmetry energy is investigated. The
kaon-nucleon interactions are described by a chiral lagrangian. Nuclear matter
energy is parametrized in terms of the isoscalar contribution and the nuclear
symmetry energy in the isovector sector. High density behaviour of nuclear
symmetry energy plays an essential role in determining the composition of the
kaon-condensed neutron star matter which in turn affects the cooling
properties. We find that the symmetry energy which decreases at higher
densities makes the kaon-condensed neutron star matter fully protonized. This
effect inhibits strongly direct URCA processes resulting in slower cooling of
neutron stars as only kaon-induced URCA cycles are present. In contrast, for
increasing symmetry energy direct URCA processes are allowed in the almost
whole density range where the kaon condensation exists.
