We have presented a method for an exact calculation of the two lowest-order, cubic (F3) and quartic (F4), perturbation terms in the Helmholtz free energy (F) of an anharmonic crystal in the high temperature limit. The method is applicable to a nearest-neighbor central force model of a monatomic fcc crystal for any phenomenological two-body potential q~(r). The calculation of F 3 and F4 requires a knowledge of the six dimensionless Brillouin zone (BZ) sums, as a function of a parameter a~ depending on the first and second derivatives of q~(r). These sums are calculated to a high degree of accuracy for a mesh size of 308,000 points in the whole BZ in the range -0.1 _< a~ _< + 0.1 in steps of 0.02. The linear temperature dependent anharmonic contribution to the specific heat at constant volume, calculated for the elements Pb, Ag, Ni, Cu, A1, Ca, and Sr, from the Morse and Rydberg potentials, is found to be positive in all cases, with the exception of Pb. In this case the Morse potential gives a negative sign. The predictions of theory are in agreement with experiments where the data is available (e.g., Cu, AI, and Pb).