What phenomenon is Hawking radiation intended to explain? I understand that science exists to make predictions, and that "unitarity" (the 100% total of the probabilities of all possible outcomes) is a major factor in the morale of scientists.  
However, in a "temporal multiverse" comprised of local universes that are no longer in causal contact with each other, I'm not sure that Hawking radiation would be essential in maintaining unitarity, since the heat necessarily radiated could be radiated into incipient "local universes" being formed within regions (black holes) that had accumulated radiation and matter prior to the disconnection of the causality within them from our side of the apparent horizon that would form when the escape velocity from them would've reached our speed of light in vacuum (which, given the fact that any BH comprising one of the aforementioned regions would have relatively dense contents, would be much higher than theirs would subsequently appear to be). 
In an environment at least marginally eternal to the past, such as Nikodem Poplawski's "Cosmology with torsion" or Aguirre and Gratton's "Steady state eternal inflation" (vetted by Vilenkin in "Arrows of time and the beginning of the universe"), it appears to me that any outgoing radiation might be balanced by radiation absorbed from its surroundings, so that the evaporation of BHs might have little effect on the computation of probabilities, even in civilizations adept at collecting and analyzing stray radiation from intergalactic voids. 
I haven't been able to find a motivation for the postulation of Hawking radiation except as a hypothesis which appears to sustain the consistency of quantum mechanics.  I might be making a number of wrong assumptions here, so I would appreciate it if the readers could help me by pointing them out.
 A: You seem to be saying, and asking if the numbers might work out, that maybe the Hawking radiation from the black hole (BH) which when contracting towards a near singularity bounces from the spin-torsion repulsive effect in high densities could be the particle production basis for the expanding universe.
Your question may be something else but that's what I gather from your last comment (which BTW is not the best way to clarify a question, it's best to actually clarify it in the body of the question).
That question, where you are hypothesizing something, and 'wondering' if true, is a way to have others do your research. That is not what this site is. But I will answer it. The Hawking radiation is that from right outside the horizon of the BH, NOT from anywhere inside it. It happens because of the horizon, and is similar to that similar radiation due to other kinds of horizons, like that due to acceleration in Rindler spacetime. It does not happen near the singularity. As @JMLCarter said, Hawking radiation arises from quantum fields outside the horizon, and Hawking showed it was so using quantum field theory in a curved spacetime, meanwhile showing how that could be calculated and be consistent with theory, without having to consider gravity (or the curved spacetime) as a dynamical field. You cannot apply the same technique near the impending singularity as gravity is very much a dynamic field. You'd have to consider the changes to the spacetime, and in the process a full theory of quantum gravity. There is not a way found to do that, we do not have a quantum gravity theory. 
Yes, you'd have to account for the particle production model, mainly for fermions as the spin arises from them, and in trying to do so in the strong gravity there you'd have to develop a quantum gravity theory. 
The arxiv reference you gave is an interesting paper, but notice that they don't get to the point of doing the quantum field theory in a rapidly changing spacetime. Their arguments and calculations simply get to the point where they get an equivalent inflation (and inflaton) model that provide the same or similar results to be equivalent to the inflation parameters measured by the Planck collaboration and other measurements, roughly. They don't show, in this arxiv paper something to fully describe their calculations, but do show some numerical results claiming the consistency with cosmological parameters. They postulate GR with torsion, using the ECSK theory, and claim to show (I did not read it close enough to verify what they said, but it is conceivable) that it is equivalent. 
They do not do quantum calculations to try to match the needed particle production rate un such an expanding BH, as a model for the Big Bounce universe. And Hawking won't help.   
A: 
What phenomenon is Hawking radiation intended to explain?

The context which it was discovered by Bekenstein is to answer the question where does the entropy disappear when a black hole forms. He predicted that black holes should have an entropy associated with them. Here, it was to maintain consistency with thermodynamics that it was discovered. 
Now, an object with entopy will have an associated temperature, which means that it ought to radiate; this obviously goes against what is understood traditionally about black holes - that they don't radiate; Hawking attempted to prove Bekenstein wrong, but in fact showed that Black holes do radiate - the eponymous Hawking radiation.   
It has nothing to do with multiverses.
