Temperature-dependent corrections to energy levels, including relativistic effects, for the positronium atom are discussed. The theoretical description of the thermal environment impact on atomic characteristics is carried out within the framework of rigorous relativistic quantum electrodynamics. As a result, the finite temperature corrections to the fine and hyperfine structure of positronium levels are evaluated. In addition, the annihilation rates of a positronium atom placed in an equilibrium thermal environment (blackbody radiation field) are studied. The numerical results are discussed throughout the paper in view of modern experiments.