Something that annoys me:

I send my pet, that gathered it's energy for 10 hours to the arena. 3 minutes later it's back, because someone attacked it 10 times in a matter of 10 seconds.

So here my idea:

if a pet wins against a pet in the arena, it has to wait a minute before attacking this specific pet again. That way people can't just spam the button for easy xp.

I get that people want to train their pets, but this would help spread the loss of energy over more pets.

I do not support this.

The situation you described is just the risk of sending the pet to the arena instead of actively picking it's battles.

Also this would rapidly become frustrating for people who are generally active at times when there are only a few pets in the arena.

I concur.
The lack of pets in my timezone is already bad enough. For instance, it's pretty much impossible to get on the weekly chart when there's nothing to fight. 

And this would definitely make it worse.