Does solid aluminum oxide conduct oxide ions at high temperatures?
Alumina is one of the most abundant compounds on Earth. If it would be also a solid electrolyte for oxygen ions, why in the world would anyone go into the trouble of dealing with the whole $\ce{Y2O3/ZrO2}$ thing? No, alumina doesn't and can't behave like that. Its oxygen ions are held in place pretty tightly.

Some forms of alumina may serve as conductors of $\ce{Na+}$ and similar cations, but that's another story.