  Spin-orbit coupling is predicted to have dramatic effects on thermal
properties of a two-component atomic Bose gas. We show that in three spatial
dimensions it lowers the critical temperature of condensation and enhances
thermal depletion of the condensate fraction. In two dimensions we show that
spin-orbit coupling destroys superfluidity at any finite temperature, modifying
dramatically the cerebrated Berezinskii-Kosterlitz-Thouless scenario. We
explain this by the increase of the number of low energy states induced by
spin-orbit coupling, enhancing the role of quantum fluctuations.
