  We consider a homogeneous Bose gas in the Gross-Pitaevskii limit at
temperatures that are comparable to the critical temperature for Bose-Einstein
condensation in the ideal gas. Our main result is an upper bound for the grand
canonical free energy in terms of two new contributions: (a) the free energy of
the interacting condensate is given in terms of an effective theory describing
its particle number fluctuations, (b) the free energy of the thermally excited
particles equals that of a temperature-dependent Bogoliubov Hamiltonian.
