It is possible to extend SC1 theory by including higherorder corrections to the free energy. We can first include the e↵ect of volume expansion through the use of the anharmonic free energy , i.e. coupling of the free energy to lattice degrees of freedom. Secondly, we can include the next order in the free energy expansion-the so-called bubble correction-which is determined from third-order force constants . As seen from Fig. , volume expansion leads to an increased softening of the mode, but the phase transition is still predicted to happen below 1000 K. Rather, inclusion of third-order force constants in phonon quasi-particle (QP) theory does result in the prediction of a soft-mode phase transition at high tem- perature (Fig. ). Fig. shows the transition temperature predicted with SC1 and various levels of QP theory at both 0 K DFT volumes and volumes predicted from anharmonic free energies. The transition temperature predicted from QP-NL theory at the 0 K DFT volume, ⇠2850 K, is in good agreement with the experimental values of 2650 K , whereas inclusion of thermal expansion increases the predicted transition temperature to above the experimental melting point. A similar behaviour was observed by Tadano et al. for CsPbBr 3 and was attributed to deficiencies in the underlying PBEsol exchange-correlation functional . Thus, anharmonic phonon theory allows for prediction of the cubic-to-tetragonal phase transition upon cooling, though with a significant dependence of the transition temperature on the level of approximation.