Dyson's eternal intelligence concept (the Dyson Scenario) states that intelligent beings would be able to think an infinite number of thoughts in an open universe.

The intelligent beings would begin by storing a finite amount of energy. They then use half (or any fraction) of this energy to power their thought. When the energy gradient created by unleashing this fraction of the stored fuel was exhausted, the beings would enter a state of zero-energy-consumption until the universe cooled. Once the universe had cooled sufficiently, half of the remaining half (one quarter of the original energy) of the intelligent beings' fuel reserves would once again be released, powering a brief period of thought once more. This would continue, with smaller and smaller amounts of energy being released. As the universe cooled, the thoughts would be slower and slower, but there would still be an infinite number of them. The idea was published in a scientific paper[1] and in a popular book[2] by Freeman Dyson.

Two recent observations have presented problems for Dyson's scenario. The first is that the expansion of the universe appears to be accelerating rather than decelerating due to a positive cosmological constant, implying that any two regions of the universe will eventually become permanently separated from one another. The second is that there appears to be a lower bound for the temperature of a vacuum, meaning that the universe would not continue to cool indefinitely .

Also, many grand unification theories predict that protons are unstable, albeit with a very long half-life. Thus the material base for intelligence could eventually disappear (in which case another type of matter could possibly be utilized). No evidence for proton decay has yet been detected, however.[3]

However, even if intelligence cannot continue its own survival indefinitely in an ever-expanding Universe, it may be able to create a `baby universe' via a wormhole in spacetime and hope that life would eventually replicate itself there.[4] Alternatively, futurist and inventor Ray Kurzweil predicts that a vastly superior race of posthumans or artificial intelligence may be able to evolve to the point when they can control the Universe, thus invalidating the heat death and continuing forever.

Freeman J. Dyson, "Time without end: Physics and biology in an open universe," Reviews of Modern Physics, Vol. 51, Issue 3 (July 1979), pp. 447-460; doi:10.1103/RevModPhys.51.447. See also here and here.
Freeman Dyson, Disturbing the Universe, 1979, ISBN 0-06-011108-9.
C. Amsler et al. (2008). "Review of Particle Physics". Physics Letters B 667: 1. Bibcode:2008PhLB..667....1P. doi:10.1016/j.physletb.2008.07.018.

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