As always, I apologise if this is well trodden ground. Feel free to ignore me or post me a link to your favourite prior thread on this topic.

Tempi is portrayed as somewhat simple. I am not entirely convinced.
Tempi is/should be appalled by Kvothe's use of power when fighting the bandit encampment.

Yet, a short time later he begins teaching Kvothe the Lethani simply because he is irritated by Kvothe's mistakes?
I would think his memories of Kvothe cutting into corpses would still be very fresh.

The following is complete speculation, and the worst kind of WAGgery, but what if Tempi wasn't so dumb?

What if Tempi sees this young man, whom he has affection for, this young man with so much potential for good, and for bad? He sees him and worries.
Tempi sees this young man and decides he needs the Lethani. And if he can't be taught that which is right then he must be controlled.

Kvothe doesn't know what Tempi told the others about him.

It's a neat trial and trap.
If Kvothe didn't come to Ademre, would they have really let him go his own way?
But every step forward is a bigger trial.

It's also possible he recognised Cinder for what he was and wanted to join Kvothe to the Adem cause...

WAGgery: to indulge in Wild Ass(ed) Guessing.
Usage: There's no evidence of that, that's just a WAG on your part.