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Since no one said, they answered this here:

http://www.enworld.org/forum/content.php?1900-D-D-5th-Edition-Sage-Advice-from-Designers-Mearls-Crawford#.VfqycvlVhBc

"Can stabilized PC spend HD after 1hr short rest even though it regain 1HP after 1d4hr? believe that is correct - at 0 hp, a short rest lets you spend HD. -M"

So according to them you could benefit from a short rest while unconscious.