Does the universe emit Hawking radiation? Would there be any experimental validation from cosmological observation?
 A: Hawking radiation is specifically associated with the presence of an event horizon. If we ignore dark energy for the moment then there is no horizon associated with an FLRW universe, so any such universe will not emit Hawking radiation.
Including dark energy changes the situation because in a flat or open universe dark energy will eventually cause the expansion to accelerate, and the universe will asymptotically approach the de Sitter geometry. A de Sitter universe has a cosmological event horizon, and this will indeed emit Hawking radiation. As far as we can tell this applies to our universe, so if we wait long enough (100 billion years or so) our universe will develop an event horizon and start emitting Hawking radiation.
You specifically mention a closed universe, but a closed geometry would not have a cosmological event horizon and therefore wouldn't produce any Hawking radiation.
The question of experimental tests seems a bit moot given that such tests won't measure anything for several times the current age of the universe.
