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Deutsch writes that even if the universe is finite (and he also discusses in the book that it can be infinite), regardless of whether the universe is heading to Big Crunch or Big Chill, it's possible to perform an infinite amount of coherent computation "until the end". Therefore, an infinite amount of knowledge can be acquired through this computation.
I can see how infinite computation can be performed in the Big Chill scenario (which is the most probable one, scientists think). In this case, cosmic-scale structures will acquire knowledge by exchanging information with ever growing latencies: years, then millions of years, then millions of millions of years, etc. However, for such structures, "knowledge" likely means something very different that it means for us. As Stephen Wolfram wrote:
In thinking about our “place in the universe” there’s also another important effect: our brains are small and slow enough that they’re not limited by the speed of light, which is why it’s possible for them to “form coherent thoughts” in the first place. If our brains were the size of planets, it would necessarily take far longer than milliseconds to “come to equilibrium”, so if we insisted on operating on those timescales there’d be no way—at least “from the outside”—to ensure a consistent thread of experience.
From “inside”, though, a planet-size brain might simply assume that it has a consistent thread of experience. And in doing this it would in a sense try to force a different physics on the universe. Would it work? Based on what we currently know, not without at least significantly changing the notions of space and time that we use.
However, I don't see how infinite amount of computation would be possible in the Big Crunch scenario (unless the universe is infinite itself). At some point when, when the diameter of the universe will be about the Plank length (or even probably much larger) the universe will not be able to perform any more computation. For infinite computation to work out in the Big Crunch scenario, the physics should have supported infinitely divisible space and matter and information carriers, Achilles-and-tortoise style, which is not the case.
Deutsch criticises finitism: "Finitism just prevents us from understanding entities beyond our direct experience." As you can see, Deutsch connects infinity and knowledge by substantiating them with one another.
Deutsch's argument about finitism is an idealistic, almost a theistic statement. Compare: "Atheism just prevents us from understanding entities beyond our direct experience." As far as I understand, this is Georg Cantor who started to think deeply about the relationship between God and infinity.
As Deutsch is also a realist, his philosophy is an interesting mix of realism and idealism, positions which are usually considered opposite to one another. Fred M Beshears arrives at the same conclusion from a different angle:
Although David Deutsch is a follower of Popper, his own philosophy is more of a synthesis of Idealism and Realism. In particular, Deutsch argues that some abstractions - those that are referred to by our best explanation of some field - should be considered to be 'objectively' real even though they are not 'physically' real. So, Deutsch does not think that ideas are less fundamentally real than external physical entities, or vice versa. Both are fundamental.
Notice that the title of this book is The Beginning of Infinity: explanations that transform the world. A hard core realist would say that coming up with a better explanation may transform our understanding of the world, but it wouldn't transform the world itself. So, it doesn't sound like Deutsch is a hard core realist. But, he's not a hard core idealist either.
I think that infinity, of course, does exist as an abstraction, a model, and it can be a useful part of other models. However, I think Deutsch is wrong when he states that universe, computation, growth of knowledge are infinite. We can apply his own attitude towards theories: they are all wrong, and we don't know what discoveries await us in the future. For example, if physicist will confirm proton decay, then infinite computation might not be possible in any universe.
An extreme application of this principle is that if we live in a simulation, it can be stopped abruptly "without any notice". Hence, even if according to the laws of physics the universe could exist forever, we can never be sure it indeed will.
Contra: