Opinions on breaking up when living together?

This will be the second break up we've had in the last year. We've been together 3, living together 2 in my house (I own it). He's insisting I give him at least a month, maybe longer to save for a new rental of his own. I think no contact would be best and he should stay at a friends house if possible. He can leave anything he needs to here until he finds a place of his own but takes what he needs and stays at a friends until then.
We are breaking up because I don't trust him. He cheated on me twice, once only 2 months in which I forgave like an idiot and the second time it was rumors from a mutual friend of the girl he was supposedly trying to date on the side. I didn't have proof at the time but have since got undeniable proof and he is still denying it.
He has already downloaded tinder on his phone less than 24 hours later. The last time we broke up (4 months ago) he would stay out until 4 in the morning several days a week so seeing that he has tinder on his phone now I can imagine what he was really doing the first time.
I just want to move on and I don't want to know what he's doing. He can't even wait to move out before he starts looking for his next girlfriend or FB so I think its fair to ask him to stay somewhere else? Or am I being unfair
Opinions on breaking up when living together?
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