I also got reminded about the method to send data from one computer to another over low frequency sound https://www.extremetech.com/computing/171949-new-type-of-aud...
https://arstechnica.com/information-technology/2016/08/new-a...
But the system still has some connection to the outside world, right? That means we could run some heavy GPU load and measure the variation in its power consumption, which apparently has been tried before: https://www.helpnetsecurity.com/2018/04/13/data-exfiltration...
Along these lines, the excess heat has to go somewhere, so maybe one could measure the variation in the work of the coolant system. I couldn't find any research about it right away (BitWhisper is similar, but a bit different), but I trust someone has already tried that.