Does it say anywhere?
It's left purposely vague because of the gap. Could be a few centuries, could be a few thousand years.
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I’m just going to estimate. Given that Golarion typically seems to develop at a slower rate than Earth does, I estimated that I’d take them a bit of time to industrialize. But gIven they’re already at rennaissance conditions, I’d estimate jo longer than 800 years before they reach current technological atabdards. Then about another 1000 before they reach full spaceflight capability. I’d say about 2000 years, but for my campaign I’m being generous and saying a solid 4000. So, in a sense, in my own personal campaign, the AR date is 8718. That is not supported by the canon anywhere as far as I’m aware. Like Garretmander said, it’s left deliberately vague.
Intentionally unknown. The gap could have been any length of time.
The tables associated with the Scepter of Ages in Artifacts and Legends actually give us a clue:
Near future 4,712 ar to 9,999 ar Modern day with political changes
Distant future 10,000 ar to 99,999 ar New cultures, new technologies
Far future 100,000 ar to 999,999 ar Totally alien cultures and science
The "distant future" entry seems to be the one that best matches with the Starfinder era, so we are looking at thousands to tens of thousands of years into Pathfinder's future.