Oh sorry, you’re right. It’s not exactly my problem.

I tested a little further. Exactly 5 minutes after closing a program (Logic or Garageband or Audio Midi Setup) the Xkey loses its BT connectivity.
It is not triggered by inactivity. For example, I can open Audio Midi Setup for hours without activity and Xkey stays connected. But if I close Audio Midi Setup, the Xkey will disconnect after 5 minutes exactly. So it’s some setting ( Mac OS X or Xkey ) that triggers the automatic shutdown when not in use.