Yes, the regulator will run cooler with an input of 9V instead of 12V. With 12V input and pulling, say, 200mA the regulator dissipates (12-5)*0.2=1.4W and with 9V input it's (9-5)*0.2=0.8W. If the regulator will overheat or not depends on the actual current and the ambient temperature. A standard 9V battery may not be able to supply the current needed for the servo though and in any case it won't last for very long.

/Henrik.