This question comes out of curiosity to find the arguments of those who make both sides of the word "or". If this is a false dichotomy I would like to hear the other options to this as well.
A few atheists have been making a similar argument to "the universe is infinite" in this very forum when discussing the Kalam Cosmological Argument, which is the reason for this thought experiment if I may put it that way. Now before anyone derails the thread saying "this is a strawman" let me make it clear that this is not an atheists position in general, but a few do make this positive claim, thus what are the philosophical or/and scientific reasonings for this?
One practical problem with an infinite universe and a cyclic universe is connected to the second law. The second law states that the entropy of the universe has to increase. While an increase in entropy will absorb energy.
For the second law to be satisfied, universally, the usable energy of the universe has to be decreasing over time, since energy needs to go into entropy, which itself has to increase over time. Energy is being bled off into increasing entropy.
The energy being constantly bled off and lost by the universe, into entropy, will be conserved, but not in a net reusable way by the material universe, since the entropy of the universe has to net increase. The universe is not a perpetual motion machine as many seem to claim. The second law suggests a one time occurrence with a finite self life. The universe is like a living thing that ages due to the second law.
The question then becomes, what is nature of the energy stored within entropy? We know that it is not net reusable by the universe, due to the second law. Entropy is considered a state variable meaning any given state of matter has a specific entropy value that is measurable. Water at 25C and 1 Atmosphere, has a fixed amount of entropy that is always the same. In that respect, the energy within entropy is similar to specific information about a very specific state. We can retrieve some of energy in entropy, by simply changing the state; go to water at 20C and 1 atmosphere. However, we cannot get the entropy energy back while also leaving the state unchanged.
The living state, in general, and the brain, in particular ,generate a lot of entropy. The energy within that entropy, if conserved, would mean your essence or your many states of life and being; soul, are conserved by the universe within the universal entropy pool. The energy for that data storage come at the expense of the universe aging.