I changed became to got. "Became cold" is correct and makes sense, but "got cold" sounds more natural and idiomatic. "Became cold" sounds a bit too literary.

@Johnsin I suppose its a matter of preference, or possibly one of those "where in the world / which variety of English" things, but I would say 'got cold' sounds very strange to my ears and that 'became cold' is what I would have naturally said. But you are not wrong, both 'got cold' and 'became cold' work.