What would be the "n" in that Big O notation, though?
If you're saying that you want accuracy out to n digits, then there are algorithms with specific complexities for calculating those. But that's still just an approximation, so those aren't any better than the real-world implementation method of simply looking up that constant rather than calculating it anew.
Put another way, 10 digits means that your error will be caused by your imprecise model of the Earth's shape, rather than imprecision in the value of pi.