  The prediction of new high entropy oxides (HEOs) remains a profound challenge
due to their inherent chemical complexity. In this work, we combine
experimental and computational methods to search for new HEOs in the
tetravalent $A$O$_2$ family, using exclusively $d^0$ and $d^{10}$ cations, and
to explain the observed phase stability of the $\alpha$-PbO$_2$ structure, as
found for the medium entropy oxide (Ti, Zr, Hf, Sn)O$_2$. Using a pairwise
approach to approximate the mixing enthalpy, we confirm that $\alpha$-PbO$_2$
is the expected lowest energy structure for this material above other
candidates including rutile, baddeleyite, and fluorite structures. We also show
that no other five-component compound composed of the tetravalent cations
considered here is expected to form under solid state synthesis conditions,
which we verify experimentally. Ultimately, we conclude that the flexible
geometry of the $\alpha$-PbO$_2$ structure can be used to understand its
stability among tetravalent HEOs.
