The volume of my speakers going down by itself and going back again in a minute or two

is a built-in test sequence occuring?

If you weren't already an old hand at this, I would have suggested going back to first principles and cleaning/checking connectors and volume pots, but I'm sure you are way beyond that.  Good luck with the gremlins!

use a stopwatch, time it, always the same time delay?

My first inclination is there is an issue with the new to you preamp. If you have a spare pre try it and see if the problem goes away. If it does that at least narrows down the component. If indeed the preamp, and not knowing which specific audio research preamp you own, there could be a motorized volume control that is linked to a microprocessor. My suspicion would be the microprocessor control reaching temperature and doing some odd things making the motorized volume control turn fully down then back to normal. If it is a motorized volume control see if the volume knob is actually turning when the volume goes up and down. Some AR don't have a motorized volume control and are microprocessor controlled. I think the problem lies within the physical means for changing the volume within the unit like the motor, what controls the motor, or a microprocessor is out of spec. Only a guess though and if true that would probably require a journey to a tech to iron out. Best of luck.

Sure sounds like the electronics need checked out to me as well. Good luck with them and hope repairs are reasonable. Probably worth keeping!