The term "res publica" never fell out of use. It shows up in the Justinian Code c. 500 AD. It gets picked up in Greek as "koinon" which is more or less a direct translation - the thing we have in common.

At least one Roman, Tacitus (1-2nd century AD) thought he was living under tyranny and blamed Augustus for it, but another historian, Velleius Paterculus (early 1st century AD) living under Tiberius thought the republic was going along quite fantastically. (The standard view is he was a shill spouting propaganda for Tiberius, but I don't know why we shouldn't take the guy at his word when he's discussing his own opinions). I suspect by Domitian a lot of the elite knew what was up, but so little survives it's hard to prove.

Just an aside: the use of the term "empire" here is kind of misleading. Rome had an empire from Punic 1, even though it was still a republic. The distinction between republic and empire here really signifies monarchy. I don't know who decided to call it "Republic" and "Empire" but I really wish they hadn't. It confuses the term and muddles analysis unless you are really into the subject.