Measurements of the specific heat, thermal conductivity, dielectric constant and thermal expansion of the single-crystal fast-ion conductor ${(\mathrm{Zr}{\mathrm{O}}_{2})}_{1\ensuremath{-}x}{({\mathrm{Y}}_{2}{\mathrm{O}}_{3})}_{x}$ with $x=0.10,0.16, \mathrm{and} 0.18$ exhibit behavior similar to that observed in amorphous solids and attributed to two-level states. The data can be fitted by the phenomenological tunneling states model, but at the expense of a more complex spectrum of two-level-state equilibration times than has generally been used for amorphous solids. The possible relationship of the two-level states with known defects is explored and found to be unsatisfactory.