  Once formed in a supernova explosion, a neutron star cools rapidly via
neutrino emission during the first 10^4-10^5 yr of its life-time. Here we
compute the axion emission rate from baryonic components of a star at
temperatures below their respective critical temperatures T_c for
normal-superfluid phase transition. The axion production is driven by a charge
neutral weak process, associated with Cooper pair breaking and recombination.
The requirement that the axion cooling does not overshadow the neutrino cooling
puts a lower bound on the axion decay constant f_a > 6 10^9 T_{c9}^{-1} GeV,
with T_{c9} = T_c/10^9 K. This translates into a upper bound on the axion mass
m_a < 10^{-3} T_{c9} eV.
