The cold soaked thing does sound like some rumour thats spread about your crews.
what I have seen on the 737ng, is put it on, door opens, runs through i presume a self test etc, then if you hit start a few mins later, the start is almost instant. I have seen a few guys do this.
ironically it was to save on their apu use stats that we get monitored by, but I think the computer will just count from when the switch is in on position not actually up to speed and burning fuel!
Again all speculation on my part!