Will the increasing rate of expansion of the universe eventually result in Hawking Radiation suddenly occurring everywhere? The universe is expanding at an ever increasing rate. Virtual particles are being created and then destroyed everywhere at every moment. As space expands even faster, will a time come when these virtual particles will be ripped apart before they can destroy each other, much like what happens at gravitational event horizons. Does this mean that the universe will at this point find itself suddenly full of Hawking radiation, almost as if a new "Big Bang" had occurred?
 A: Yes, cosmological horizons are expected to emit Hawking radiation. However, the temperature would be extremely low, about $10^{-30}$ K (Baez 2004 and Hu 2010). This is nothing like a Big Bang. This is all assuming dark energy that behaves like a cosmological constant. For dark energy with certain properties, you can get a Big Rip. See Adams 1997 for a more general discussion of the ultimate fate of the universe, which however was written before the discovery of dark energy. 
You may also be interested in Penrose's conformal cyclic cosmology, although the theory no longer seems to be viable.
Adams and Laughlin,  http://arxiv.org/abs/astro-ph/9701131
Baez, J., 2004, "The End of the Universe.", http://math.ucr.edu/home/baez/end.html
Hu, 2010, http://arxiv.org/abs/1007.4044
A: Have you read  this  simple enough article?

Once objects are bound by gravity, they no longer recede from each other. Thus, the Andromeda galaxy, which is bound to the Milky Way galaxy, is actually falling towards us and is not expanding away. Within our Local Group of galaxies, the gravitational interactions have changed the inertial patterns of objects such that there is no cosmological expansion taking place. Once one goes beyond the local group, the inertial expansion is measurable, though systematic gravitational effects imply that larger and larger parts of space will eventually fall out of the "Hubble Flow" and end up as bound, non-expanding objects up to the scales of superclusters of galaxies.

Bound states , including our galaxy which is gravitationally bound are infinitesimally affected by the expansion. Virtual states exist as mathematically useful models  only in bound form.  To get a real pair creation out of a virtual pair energy must be supplied so that the particle appears with mass . In current theories the expansion is not supplying energy in the local regions, where conservation laws hold .
In this article scenaria for the end of the universe are described.
Edit after the conflictin answer of Ben
You ask 

Does this mean that the universe will at this point find itself suddenly full of Hawking radiation, almost as if a new "Big Bang" had occurred?

The answer is no, because at the temperature for the model estimated in Ben's answer, even if one could talk of virtual particles , even the eV energies needed for neutrino antineutrino pairs will have infinitesimally small probability of getting energy from the tail of the statistical distribution. So it will not be filled with Hawking type radiation.
As I said in the comment, even the concept of virtual particles, inextricably  tied to Feynman diagrams, will have little meaning if there are no bound states. The whole Standard Model will be non existent ( quarks cannot be free), virtual particles and all, and Hawking radiation needs Feynman diagrams.
