  Axion-like particles are predicted in many physics scenarios beyond the
Standard Model (SM). Their interactions with SM particles may arise from the
triangle anomaly of the associated global symmetry, along with other SM global
and gauge symmetries, including anomalies with the global baryon number and
electromagnetic gauge symmetries. We initiate the phenomenological study of the
corresponding ``electrobaryonic axion", a particle that couples with both the
baryon chemical potential and the electromagnetic field. Neutron stars,
particularly magnetars, possessing high baryon density and strong magnetic
fields, can naturally develop a thin axion hair around their surface. In this
study, we calculate this phenomenon, considering the effects of neutron star
rotation and general relativity. For axion particles lighter than the neutron
star rotation frequency, the anomalous interaction can also induce the emission
of axion particles from the neutron star. This emission, in the light axion
regime, can have a significant contribution to the neutron star cooling rate.
