We carry out a complete calculation of the thermodynamic properties of Kr from a potential function, the Aziz potential, including the three-body Axilrod–Teller contribution, and the (λ4) anharmonic perturbation theory proposed by Shukla and Cowley (Phys. Rev. B: Solid State, 3, 4055 (1971)), where λ is the Van Hove ordering parameter. Along with the λ4 results, in the high-temperature (HT) limit (T > θD, where θD is the Debye temperature), we also present the results for the quasi-harmonic (QH) theory (calculated for all temperatures), the lowest order (λ2) perturbation theory (PT), as well as results from those theories that involve a subset of diagrams (contributions) of 0(λ4), both in the HT limit. This work represents the first calculation of the thermodynamic properties of Kr with the λ2 and λ4 anharmonic PT from a potential function not fitted to the crystal data. The Aziz potential gives an excellent description of phonon-dispersion curves in the three principal symmetry directions. The QH results are in good agreement with the experimental values for most of the thermodynamic quantities for temperatures up to <Tm/3 (Tm is the melting temperature) except for the isothermal bulk modulus (BT) where the agreement is poor in 0 K < T < 25 K and good up to 2Tm/3. The λ2 PT results are only slightly better than the corresponding QH results in the temperature range of < Tm/2. The inclusion of the λ4 PT enhances the results for the Aziz potential significantly. The calculated lattice parameter (a0) is in excellent agreement with experimental values up to 3Tm/4. For T > 3Tm/4, a0 rises rapidly and there is an indication of the breakdown of the perturbation expansion. The values for specific heats at constant volume (Cv) and constant pressure (Cp) and volume expansion (β) are in very good agreement with experiment up to 60% of Tm. The other schemes (with the exception of Ladder) that utilize a subset of diagrams of 0(λ4), which were so successful for the model of a Lennard–Jones solid (viz., ISC (improved self-consistent), λ4-Ladder etc.), are not so useful for this potential. This is due to the heavy cancellation of diagrams in these sets. The ring diagram scheme proposed here for the Aziz potential gives better results than the ISC scheme.