Two billion years' work is a hefty amount in terms of scientific and technological development, assuming people survive. So, as David Deutsch would point out, what actually happens will depend on what solar engineers choose to do.
Assuming the technology is there, we will have lots of choices available.
Maybe we can even do both.
This is a rather large assumption to make, considering we can’t even control the environment of our own planet currently.
It can happen that any crisis wipes us, or even life in the planet, out.
Stars have to be above a particular weight to go supernova - the sun will grow into a red giant, and then, once it burns most of it's hydrogen, collapse into a brown dwarf.
-- Karl Popper
Loves this one for example:
15,000 yrs from now: According to the Sahara pump theory, the precession of Earth's poles will move the North African Monsoon far enough north to convert the Sahara back into a tropical climate, as it was 5,000–10,000 years ago.