Apartment requesting security deposit before full approval?

I'm currently renting a house and have never rented an apartment before, but I'm having to cut back on expenses. When signing the lease for my house, I was approved and had to pay my security deposit but I was also given the lease agreement to read and sign.

Now, I'm applying for an apartment and they're requesting the security deposit before finalizing my rental application approval. Apparently, after paying the $50 application fee I'm "pre-approved." I haven't been given a lease to sign. This seems weird to me and I'm not sure if I should continue with this application or not. Is this normal for Houston?