I understand that in 500m years luminosity will increase by 5%. By then we should be able to survive that modest increase. Where did you get 500m years?
And when you consider that by this point energy input is on a permanent increase, there's really not much you can do apart from moving the whole damn planet out of the way.
It's kind of pointless to speculate what kind of technology we'll have in 500m years (or indeed if there is a 'we' left in the system). The time scale is so enormous that there's simply no way to predict anything at all involving humans or human-derivatives.
The planet (without literally unimaginable intervention) will heat up and become uninhabitable sometime around 500m years from now.
You can put something in between, a giant sun shade
Maybe someone smarter than me can calculate the optimal parameters to get the smallest shade that orbits the sun. If we are looking to only block out a percentage of the sun then it doesn’t even need to be fully opaque, maybe a giant dream catcher would be fine.
And already a known option for our current global warming situation.
That means you're heavily restricted in where you can put it. It has to be either right next to earth, in which case it's hard to keep away from Earth, or at L1, which is not stable. There's no other place you can put something where it will stay directly between us and the sun.
I suspect the atmosphere, ecology and albedo would adjust to radiate the extra energy away anyway.
Now I'm actually morbidly curious: do you think humans could even come close to sterilizing this planet if we wanted to? I mean maybe we could burn away the atmosphere if we set off enough nukes (probably more than humanity's ready-to-use-stockpile, I think?), but the biomass that's inside the ocean (...would that even remain without an atmosphere or would it boil off? regardless...) and crust is simply massive. I strongly suspect that life would keep on going.
That’s a really misleading way to describe what going on. The amount of energy radiated into space is a function of temperature, so increasing incoming radiation also increases outgoing radiation. 5% increase in luminosity is roughly 3.5C increase in earths average surface temperatures ignoring other effects.
However, you get a feedback loop as water vapor is a greenhouse gas which makes things hard to model. Here’s a study suggesting the tipping point is on 1 billion years https://www.sciencedaily.com/releases/2013/12/131216142310.h... but it’s not suggesting the oceans literally boil but rather a transition point where there’s a lot more water in the atmosphere.