  We study the stability of a spherically symmetric black hole with a global
monopole hair. Asymptotically the spacetime is flat but has a deficit solid
angle which depends on the vacuum expectation value of the scalar field. When
the vacuum expectation value is larger than a certain critical value, this
spacetime has a cosmological event horizon. We investigate the stability of
these solutions against the spherical and polar perturbations and confirm that
the global monopole hair is stable in both cases. Although we consider some
particular modes in the polar case, our analysis suggests the conservation of
the "topological charge" in the presence of the event horizons and violation of
black hole no-hair conjecture in asymptotically non-flat spacetime.
