Landlords We Need Our Pets

I have been trying desperately to find a decent rental property but when I tell them I have an emotional support pet and that he is a cat before I can get any more words out they shut me down cold. They are rude, mouthy and intolerable.
My cat is 3 yrs old, up to date on all his shots, I take very good care of him and I clean after him all the time. Yet the landlords won't even let me get any of that out. They tell me there is no such thing and to give up my pet to the pound if I want a nice place to live.