  Nuclear medium effects in the neutrino cooling of neutron stars through the
exotic reaction channel \gamma \gamma --> \pi^0--> \nu_R \bar{\nu_L} (\nu_L
\bar{\nu_R}) are incorporated. Throughout the paper we discuss different
possibilities of right-handed neutrinos, massive left-handed neutrinos and
standard massless left-handed neutrinos (reaction is then allowed only with
medium modified vertices). It is demonstrated that multi-particle effects
suppress the rate of this reaction channel by 6-7 orders of magnitude that does
not allow to decrease existing experimental upper limit on the corresponding
\pi^0\nu\bar{\nu} coupling. Other possibilities of the manifestation of the
given reaction channel in differente physical situations, e.g. in the quark
color superconducting cores of some neutron stars, are also discussed. We
demonstrate that in the color-flavor-locked superconducting phase for
temperatures T < (0.1-10) MeV (depending on the effective pion mass and the
decay width) the process is feasibly the most efficient neutrino cooling
process, although the absolute value of the reaction is rather small.
