The thermodynamic potential of quantum chromodynamics is calculated in orders 1, αc and αc 3 2. The renormalization group is used to improve the expansion by allowing αc to be a function of the temperature and chemical potentials. The results are valid for an arbitrary number of quark flavors and masses. Numerical calculations are made for the special case of up, down and strange quarks. Particular attention is paid to the breakdown of the perturbation expansion, and possible connection to the thermodynamics of Hagedorn is discussed. The MIT bag model, which has a fixed coupling constant, does not appear to yield good convergence at high temperatures.