[LEV = Longevity Escape Velocity, the idea that research will be able to extend lives faster than existing people are aging].
If it's expensive and rare, nothing will change. If it's cheap and common, everything will change, and what happens to criminals who have no appeal chances is one of the least important aspects of it.
My genuine best guess is "they don't, actually, get offered longevity extension; prisoners can't even expect to get life-saving prescribed medicine or obviously indicated treatments (eg to get a broken leg splinted), let alone HRT or anything elective; also, approximately no one has incentive to let prisoners get (presumably expensive) longevity extension therapy, unless it's to make damn sure they serve every last one of those 30,000 years."
See I don’t expect LEV tier longevity treatments to possibly exist before AGI so I also don’t expect it to be expensive if it does exist.
The meaning of "life sentence" appears to vary wildly from one jurisdiction to another. Anywhere that specifies a duration, people would probably just serve out their time. Elsewhere, some new law would have to get written in light of the new human lifespan.
How much does the rest of the world change?
Suppose that things in general are being run by pervasive AI that monitors everything, with every human being watched by many humans-worth of intelligence, and fast enough, ubiquitous enough robotics to stop most or all human actions before they can be completed. Why would you even have prison sentences of any kind?
If you hold everything constant and just vastly extend everybody's life span, then maybe they stay in prison until it becomes unfashionable to be so punitive, and then get released. Which doesn't mean that kind of punitiveness won't come back into fashion later. Attitudes like that can change a lot in a few centuries. For that matter the governments that enforce the rules have a shelf life.
Yes prison sentences don't make sense after the fact. To be clear, I mean existing life sentences.
For those, I think it depends on how much the rule deciders decide to uphold laws. To let them out requires rewriting current laws and retroactively applying them. If the the rule decider generally follows democratic will eventually they will be let out.
OTOH, if they vaguely uphold the status quo and rigidly follow a constitution, I can see certain people locked up forever. Some prison sentences are written in years and many life sentences still have a time limit. In reality, some old criminals under a life sentence are also let out. I can see immortal entities under existing life sentences arguing their punishment is cruel and unusual and eventually succeeding.
I also meant existing life sentences. At any given time, you may have a political change that ends them, and once that happens, it's as much a matter of law as the original sentence.
I can't see any given set of laws or constitution, or the sentences imposed under them, lasting more than a few hundred years, and probably much less.
I could see a world where they didn't get the treatments to begin with, though.
Presumably they get offered longevity treatments since they already get healthcare. Are they locked up until the end of time? For 100 years?