The AI will sacrifice Kami of False Hope before combat.

A specific situation was this: the opponent was mono-white and at low life. I cast a creature with protection from white before combat. The AI sacrificed the Kami to fog immediately. My creature couldn't attack this turn so the fog did nothing. If the AI was trying to protect itself from a lethal attack, which I suppose it what happened, it might also mean that it has trouble understanding summoning sickness on enemy threats.

In any case, timing should be either in response to enemy removal if it would benefit from it, or otherwise after the opponent declares attackers.