Entropic stabilisation of cubic ZrO 2 . Since the anharmonic free energies fail to describe the phase transition, we posit that other entropic factors must stabilise the cubic phase. Given the consistent 20-40 meV atom 1 greater stability of the tetragonal phase, an additional entropic stabilisation, S other in Eq. 1, on the order of ⇠0.01 meV atom 1 K 1 is needed at the phase transition temperature. We consider three origins: (i) higher-order anharmonic contributions, e.g. the temperature dependent internal coordinates are not included at the present level of theory ; (ii) a dynamic or static ensemble of local tetragonal domains to produce a quasi-cubic phase on average; (iii) fast ionic conductivity, which is a common feature of fluorite type structures . These possibilities can be tested through various aspects of ab initio molecular dynamics (AIMD) simulations. 1. Higher-order anharmonicity. This contribution is probed using constrained MD with the Blue Moon ensemble along an order parameter given as the collective oxygen displacement along the soft-mode at the X-point . This gives access to the free energy gradient along the order parameter (see Methods) which can be integrated to obtain the free energy as a function of the order parameter. As shown in Fig. , the free energy surface remains of double well nature at temperatures up to 3000 K. Thus, no transition is predicted to occur below the melting point. Thus, we cannot attribute the stabilisation of cubic zirconia to higher orders of anharmonicity not included with the current level of SCPH. Importantly, with the constrained MD, no ionic di↵usion is allowed.