Some Android devices turn off the USB port when the screen turns off, which is only preventable by an application telling Android to stay awake using a wake-lock (or by keeping the screen on to avoid the situation).
It is unlikely Endomondo would add support for this feature, but you could always check with them too. There are some apps which can make the screen incredibly dim to save power but still keep it on, or you might find one that can enable a wake-lock to prevent the system and USB from going to sleep.
