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Based on the correlative method of an unsymmetrized self-consistent field,16–23 we have derived expressions for elastic constant tensors of strongly anharmonic crystals of cubic symmetry. Each isothermal elastic constant consists of four terms. The first one is the zeroth approximation containing the main anharmonicity (up to the fourth order). The second term is the quantum correction. It is important at temperatures below the De-bye characteristic temperature. Finally, the third and fourth terms are the perturbation theory corrections which take into account the influence of the correlations in atomic displacements from the lattice points and that of the high-order anharmonicity respectively. These corrections appear to be small up to the melting temperatures. It is sufficient for a personal computer to perform all our calculations with just a little computer time. A comparison with certain Monte Carlo simulations and with experimental data for Ar and Kr is made. For the most part, our results are between. The quasi-harmonic approximation fails at high temperatures, confirming once again the crucial role of strong anharmonicity.