Approved for place, but landlord holding off on signing lease. Can they increase rent?

My roommate and I recently found a place and were approved for it by the rental company. While I understand different companies have different policies, I am concerned because the company doesn't want us to sign the lease until a few days before we move in on the 15th. In our rental application we agreed to a certain months rent and paid that amount in full as a security deposit. My question is, without signing a lease, can they raise that amount in the next 9 days if it was not agreed on previously but we never signed for that amount? Does the application and security deposit secure that amount and allow us any legal rights in refusing to pay if they increase the rental amount? Do we lose the place if we fight for that original amount? We're concerned about updates being made to the place and the rent subsequently being increased as a result. We have to be out by the 15th and don't want to be without a place to stay because of this.