[eru]

Though, of course, you might expect to see that, even if life can arise in extreme environments:

It's uncontroversial that life is much denser in more hospitable environments. If the extreme environments are in regular contact with the hospitable environments, and if evolution is sufficiently random, you would expect over time that most organisms anywhere have an ancestor in the hospitable environment.

Even if life can arise in the extreme environment.

[bitreality]

What is the most hospitable environment? A tide pool? Also the most hospitable environment for genesis is likely different from the most hospitable environment for a complex ecosystem.

[Symmetry]

For life without photosynthesis, which takes a while to develop, some place with a lot of free chemical energy because the first replicators will be very inefficient. White and black smokers under the sea are usually put forward as candidates.

[eru]

That's an additional good question that would complicate the simplistic model I suggested in my comment.