Amp — Leave on or turn off

This what others have already said, component dependent.  Our tubes amps change in about 20-30 minutes and can max out in 2 hours or less.

Happy Listening.

I tend to lean towards leaving things on, and digital/transistor for sure.

I think what the manufacturer recommends is irrelevant.  What matters  is your own ears, and your willingness to pay the electric bills. :D

If you can't hear a difference, leave your gear off.  If you can, leave it on.  It's that simple.  No measurement is going to tell you any better.

RV, I have repeated your experiment a number of times with the same results. On longer always sounds better, but with diminishing returns, for about a week.

Its’s true of my DIY phono/pre - it’s on all the time. It’s true of my DIY Class A amps (they idle continuously at half power). It’s true of my Bryston SST in the HT (although B recommends turning it off). But since HT doesn’t matter nearly so much, I do turn the B off, and then back on an hour before - but an hour isn’t really enough.

My cartridges take 10-40 minutes too.

As for the car analogy, I ALWAYS used to warm up my engine carefully, and my tires too, before taking a fast lap.

YMMV

I had a Krell FPB 600 amp that I always left in the standby mode until I powered it up to listen. Was doing that for years and years. Then I have now the ARC Ref 750s. And I always let them warm up about 1 hour before listening. Happy listening.