Warming up is related to, but different from running in. Warming up happens each time a component has been disconnected from the mains power
It is a phenomenon that is not accepted by everyone and that’s partly very understandable. Some components simply do not need warming up. The Meridian 508.24 CD player for example, doesn’t seem to change its character after warm up. It does have to run in briefly when purchased new, but after that I have never heard meaningful differences.
Other components simply sound really bad unless they haven’t been switched on for some time. An example is the Wadia 861 CD player that really has to spin a few discs before it starts sounding fluid and engaging. This is most likely the caused by the rigid VRDS disc clamping system inside.
Many preamps and poweramps need warmup time before they start to sound musical. When cold they can sound perfectly fine, but many really only start sounding really musical after at least 30 minutes of playing time.
CD players and preamps don’t normally consume a lot of power so therefore it’s not such a bad idea to just leave them switched on all the time so that their electronics are ready to go at any time. Some power amps consume enormous amounts of power, even when idle (ie switched on but not playing music) so even though these too benefit from warm up, it’s too costly to always leave them on. Some power amps have standby functions.
A McIntosh MA6900 integrated amp has a soft-touch powerbutton that seems to initiate a standbut is actually completely switches off the amp completely. The amp sounds controlled and powerful right after switch on but really needs at least 30 minutes to start sounding colourful and fluid, and the longer it is on, the better it sounds. It really is one of the most “continuous” sounding amps that I have encountered, but only after it has been running a while.
While sometimes the standby position is nothing more than a fully off position while a separately powered remote control circuit can switch the amp on, in some cases the standby position is implemented very cleverly. Jeff Rowland model 6 poweramps for example are fully switched on in standby, but with greatly reduced bias current and disabled speaker connectors. This allows them to sound great from the very first instant after switching on from standby, with all the detailing and subtlety that they are capable of. But even so these amps sound better after 1 hour of playing, the sound becoming rounder, and tonally fuller.
Most of the time a given component already sounds good even starting from cold, assuming it has been run in before and isn’t brand new. Exceptions aside, when a run-in component doesn’t sound appealing at all when starting from a cold state, it probably will not change its character entirely. It will improve in areas such as focus, smoothness, soundstage size and fluidity but its basic character may not change all that much.