  We study a neutron star with a quark matter core under extremely strong
magnetic fields. We investigate the possibility of an Urca process as a
mechanism for the cooling of such a star. We found that apart from very
particular cases, the Urca process cannot occur. We also study the stability of
zero sound modes under the same conditions. We derive limits for the coupling
constant of an effective theory, in order the zero sound to be undamped. We
show that zero sound modes can help kinematically to facilitate a cooling
process.
