When the new logics are connected before opening the app, the power usage is lower than after opening (and closing) the app. This is obvious because the LED is still turned on and the device is warmer.
Current before opening app: ~0.12A
Current after opening app: ~0.26A
Current after closing app: Still ~0.26A
Since the software has to configure the device again anyway on the next startup, even if the logic's already on, there would be no time loss. However, there would be significant energy savings (~50%) if the logic was turned off again after closing the software.