It could, but as someone who has spoofed user-agents in the past (primarily to get Chrome-only websites to cooperate) I would prefer if it wouldn't. If the baddies can snoop my https traffic or directly copy the auth cookies from my machine then also copying my user-agent isn't that big of a step for them. One might argue that detecting changes in user agents could be part of some kind of defense in depth strategy, but as a user I imagine I'm already so boned in that scenario that I doubt it would save me. So overall such a mechanism would bring me more inconvenience than security.
That's the whole point of RBA, though. That two requests have the same user agent doesn't tell me much, but if you have two different user agents from two different IPs that may sound really risky (use case dependent, of course).