  We study the effect of thermal fluctuations on the topological stability of
chiral p-wave superconductors. We consider two models of superconductors:
spinless and spinful with a focus on topological properties and Majorana
zero-energy modes. We show that proliferation of vortex-antivortex pairs above
the Kosterlitz-Thouless temperature T_KT drives the transition from a thermal
Quantum Hall insulator to a thermal metal/insulator, and dramatically modifies
the ground-state degeneracy splitting. Therefore, in order to utilize 2D chiral
p-wave superconductors for topological quantum computing, the temperature
should be much smaller than T_KT. Within the spinful chiral p-wave model, we
also investigate the interplay between half-quantum vortices carrying Majorana
zero-energy modes and full-quantum vortices having trivial topological charge,
and discuss topological properties of half-quantum vortices in the background
of proliferating full-quantum vortices.
