There are (as I see it) 3 issues with your concept of AGI immortality.
1) Liveness of Intelligence
The assumption that intelligence, consciousness and 'liveness' are all synonyms is a fallacy; intelligence is the capacity to identify and recognise complex patterns and because we (as humans) tend to anthropomorphise anything that we see as 'intelligent', we imbue a sense of consciousness and liveness to such an entity.
Intelligence is NOT the same thing as consciousness and as such, it is not safe to say that an intelligence, natural or artificial, is conscious, let alone alive. If we are talking an algorithmic, computer based intelligence, it is further safe to say that intelligence may well be immortal by virtue of the fact that it can be indefinitely maintained, but that is not the same thing as being alive.
2) Is immortality infinite?
As per your own comment that the universe will eventually die, immortality is an interesting concept from a mathematical perspective - if we think of the universe from the conecept of entropy (2nd law of thermodynamics), eventually the universe has to 'run out' of order and die via one of several possible end states, like the Big Crunch or the Cold Death.
As such, if one defines immortality as an infinite state of being, the answer is going to be no, an AGI cannot be immortal because the universe is not immortal. But, if we define immortal as existing through to the end of the universe (or close thereto), then we still have a possibility in that regard.
3) Strong AI (Mathematics)
If we take the concept of strong AI as an abstract theory, then that infers several attributes to your AGI;
a) The universe itself is an algorithm, and therefore fixed in its capacity to understand itself
b) your AGI can only ever (at best) be a simplified model of the algorithm that defines the universe.
Put another way, Strong AI (or formalism) basically defines the universe as a (complex) algorithm in its own right, meaning that our own consciousness is simply a function of complexity, which in turn means that our lives are fixed in a physics-compatible version of fate. If this is the case, then your AGI is merely a complex algorithm that is (by definition) a subset of the algorithm that defines the existence of the universe.
If this is the case, then with the right conditions, your AGI can be considered immortal under the confines of the universe's scope of existence - assuming that we don't linguistically define immortality as the ability to survive beyond the universe.
The catch with this thinking, is that in a formalist universe, our very consciousness and liveness is merely an illusion because any point in time in such a universe can be predicted from any other point in time, provided one perfectly understands the algorithm on which the universe is based. In other words, if one can predict the future perfectly, one cannot alter it, meaning free will is an illusion.
Conclusion and Summary
If an AGI can exist with a sense of liveness, then human liveness is algorithmically based, meaning free will is an illusion. In such a case, it's possible that an AGI may well be immortal according to the scope of the universe, but that implies that we ourselves are not capable of changing anything, or making choices of our own because the universe has already dictated our 'fates'.
On the other hand, if the universe is non-algorithmic, and by extension human liveness and consciousness is non-algorithmic, then it's possible that an AGI has the capacity to be immortal, but never be alive in the sense that we are because computers are algorithmic by nature and therefore not capable of free will as we understand it philosophically.
It's a conundrum to be sure, but ultimately it all comes down to this; if the universe is algorithmic, then an AGI can be immortal but free will is a lie. If not, then an AGI can exist into immortality, but never be alive.
It all comes down to whether or not you believe in your own sense of free will.
