Sure. But the thermal mass of an in-use pot is significant, and that buys you time. You can read more about the details here:

http://www.aluminiumtoday.com/contentimages/features/Oyeweb....

It's a well-done summary, I think you'll find it interesting. The tl;dr is that power interruptions up to ~4-5 hours do not cause significant trouble.

Huh, interesting. I read the whole thing and understood at least half of it ;).

Are the damages primarily thermal? I couldn't figure it out from that report, although it seemed to somewhat hint that they were. I.e. if you could keep the cells warm (with more insulation or otherwise), would they still get damaged by the shutdown?

Your assumption is correct. The vast majority of the damage comes from thermal stressing of the refractory materials that make up the crucible of the furnace. If there were better insulators available, so that the rate of heat loss through the furnace was minimized, it would be a huge improvement to the industry, both in energy saved during production, and greater amounts of time available after shutdown before thermal effects wreck the furnace.