I toasted two rectifier tubes doing this accidentally. I am surely not an expert but I think that this also damaged the can capacitor because the can went bad after the flashing the rectifiers from hot power cycles. For what it's worth this happened because the smart switch the amp is on malfunctioned, not the actual switch but the internet support system that smart switch runs on. My attempted fix for this worked out poorly and the rectifier flashed. At that point the can cap had an intermittent short or something and the amp was making a loud hum noise at the speaker.
I replaced the can cap and all was good. I think it was about $45 for the can cap and $30 for two rectifiers. Plus there was a lot of agita on my part because I don't know diddly about electronics, but am learning, unfortunately. I actually soldered the can cap back in myself after local shops wanted the amp for weeks with big service charges.
I may be wrong but I think turning off the amp first allows incoming music to drain down the capacitors. It does prevent that "pop" sound and, I think, ameliorates, somewhat, the hot cycle problem.
The bottom line is don't hot cycle tube amps. I have no idea why you'd want to turn an amp on and off in under a couple minutes anyhow.