For a cheap option, look for sublets. Might not be ideal living conditions, and might be sharing spaces with others, but for Jul/Aug I think many students especially are looking for people to occupy their place while they are out for summer break.

Not sure if monthly rate hotels will allow for pets. The cat is definitely the hardest thing to figure out to relocate well, since worst case you can stuff things in a self-storage unit until you can move in.