I don't know what type of caps they use in the Odyssey amps, but 500 hours of break-in is not unheard of for certain caps. SO saying this am-p requires 500 hours to completely settle in is not something that would strike me as out of bounds.
As for keeping the amp continuously powered on, I had some Audio Alchemy D/A gear back in the 90s that sounded dreadful for quite a few hours after it power up, even if it was of for as little as five minutes, so I just kept that powered on all the time. My tube gear always got powered down when not in use to conserve tube life, but being completely SS, there was no compelling reason to power down the AA gear, so I kept that powered up for years at a time.
Looking at the specs for the Odyssey, it only draws 30 watts at idle, so it is not going to have a big impact on your electric bill So if they think it will sound better left on all the time I personally would not think twice about going with that.