Will a 78S12 be cooler than a 7812 when handling the same load?
I'm comparing the datasheets for ST Microelectronics' versions of these:


L78S12CV
L7812CV


The performance characteristics, such as Output voltage vs. junction temperature, is identical for both.

The operating (junction) temperature is different, 7812 is 0-125°C; 78S12 is 0-150°C.

In both cases, in a TO-220 package, the thermal resistance is the same: 5°C/W for junction-case, and 50°C/W for junction-ambient.

Based on this information, I would not expect the 78S12 to run cooler given the same load and heatsink. But it can handle higher operating temperatures, which may be a reason to favor it.