I disagree. The B+ voltage is electronically switched on and off internally in the audio head by a serial data signal from the BCM. If that were true, the radio would lose the clock memory!
It also powers the memory and the clock full time. AFAIK, every H3 factory radio turns off when the door is opened or within 20 minutes of ignition OFF and that's RAP. The manual also states that audio is on RAP.
I think the main problem is, I'm talking totally technical and you are not. You might call it the RAP circuit, some moron contractor who wrote the Hummer owner manual might call it a RAP circuit function, but it is NOT technically a RAP function. Below you will see a RAP circuit diagram for the H3, and it shows NO RADIO on the RAP circuit. (GM manuals were written by contractors working for contract houses and were very seldom passed by GM engineering for review. They were passed through GM legal for content. Most of those moron who wrote GM manuals probably thought RAP pertained to a song.)
The power to the radio is directly from the battery. It is noted as B+ in the diagram and B+ denotes direct from the battery! If the radio was manually turned On via the switch when the H3's ignition is turned to On, the BCM signals the internal computer within the Radio to supply power from the B+ circuit to the circuitry that controls audio, display, etc. That signal will also go to the amplifier and Onstar module if so equipped.
When the ignition is turned Off, and the door is opened, or if after approx. 20 minutes, the BCM will turn off the radio by signalling the radio's computer, again via the serial data link. However, it does NOT shut down the B+ signal into the radio in order to maintain the clock setting and possibly the radio stations (can't remember if the stations are affected or are on a non-volatile memory chip).
As for the actual RAP circuit, that is controlled via a RAP relay. When the BCM supplies a ground to the RAP/ACCY relay the circuit is closed, the relay closes and B+ is supplied to the components as shown in the diagram and not one of those components say radio!
The RAP circuit is comprised of (note there are two or three circuits not shown for GM SPO accessories that can be added at a different time):
The reason this is all brought up is that since the Radio is not part of the RAP circuit, if the door is opened/closed and the interior lights go out, but the radio remains on, it is most likely not a door jamb switch problem since the door jamb switch is the main input to the BCM to shut off the radio and turn off the RAP circuit, and if the radio remains on it is most likely a fault of the BCM or like NEO jogged my memory, the internals of the radio. Sometimes, just like any computer, a reload of the software to the BCM will stop the computer farts. And when diagnosing electrical circuits you have to narrow down the problem and taking the radio away from the actual RAP circuit helps a hell of a lot.
However, I have not seen the OP come back and clear up some of the questions, such as is it a stock or aftermarket radio, do the light go out when the door is closed and the radio remains on. Without that info, it can be hard to even come close to a diagnosis.