That’s regionally specific then, because they sure as hell don’t do that where i live (EU member). They have to compare with non-sale price within a month or something, so it’s complete bullshit here because they artificially inflate prices prior to black Friday “sales”.
If its a cheap crappy one, the compressor is on/off depending on temperature. Decent ones will have a VFD to manage the load of the compressor so it doesn’t have to turn on/off all the time but just regulates the compressor load to match heating/cooling requirement. Both have their own controls, and you generally shouldn’t mess with them.