Moved to Electrical:

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I can't get to the wiring diagrams right now but many of the items in the mini are supplied with power at all times and switched on and off by low level signaling with in the computer system. This is probably what is happening. Unfortunately I do not know that much about the Sportster. It probably does not have a turn of timer, and if it did it would be turning off on you when you were listening to it.

I believe that there is a power line for the amp if you have the HiFi system that might be able to be tapped into to get it to turn off but unitl I can get to the wireing diagrams again I can not be sure.