So you want to live forever? You might - even in a finite universe:

As Dyson imagined, a sense of purpose would motivate cognizant life to try to maintain itself as long as humanly — and then transhumanly — possible. [...] As the universe continued to cool, our AI descendants would need to take action. [...] [Dyson] imagines a gradual slowing down of thinking processes. Only necessary thoughts would transpire and these would happen at an increasingly snail-like pace. Between thoughts, the AI devices would hibernate to conserve vital, usable energy. By spacing out thoughts more and more, Dyson argues, intelligent existence could persist almost indefinitely, although the number of total thoughts would still be finite.

-- From Paul Halpers summary Learning to chill of Dyson Freemans Time without End.