This is exactly what I did when I was searching for an apartment. I had a cat, and she was coming with me. My requirements for an apartment included that they allow pets and that the pet deposit/fee wasn't ridiculous.
On the other side of things, you have my friend Melissa. Melissa decided she wanted to move closer to work. She picked a more expensive apartment, and decided she couldn't afford the pet fee and therefore couldn't take her cat. So then she was posting on Facebook for somebody to please take her cat, because if she couldn't find somebody, she was going to take him to the shelter. She had only rescued him from a shelter 6 months earlier and he was a year old! So I took him in.. and I'm keeping him for the rest of his life. Her situation was 100% preventable. She didn't have to move into an apartment where she couldn't afford the pet fee. I don't understand it. He's a great cat, too.