Windows Phone 8.1 is so terrible!! It has now support for Bluetooth Low Energy, but it does NOT support gps background apps any more. This is absolutely unbelievable, but it is true, see http://msdn.microsoft.com/library/windows/apps/dn642084(v=vs.105).aspx
So ... Run.GPS cannot use the Windows Phone 8.1 API. It does run on WP8.1, but it cannot be a WP8.1 app :-)
We've invested much time to port all the Bluetooth Low Energy features for WP8.1, but when we uploaded it to Windows Marketplace, the app was simply refused because it was trying to track the locaction while in background mode :-(
The problem is that WP8 is a very sandboxed system. For instance, apps are allowed to read from the SD card, but they are not allowed to write to it. Therefore, the main data storage for Run.GPS must be the internal memory.
This also means that if you download a large map (.ratlas) file from GPS-Sport.net, this is also stored in internal memory, which is pretty stupid.
How about this solution: you upload your trainings to gps-sport.net from the old device.
Then you put the trainings in your GPS-Sport.net download cart. They will be downloaded as routes, not as trainings, but that does not matter since the timestamps are also in the routes. So you can use these to compete against.
I have found out that RunGPS on WP8 devices can only save trainings in internal memory, cannot save trainigns to SD card. This is big disadvantage: after moving to new device the RunGPS loses one important feature: virtual training agains yourself. Due to that all my trainigs are left on the old device, the trainings log has gone, I cannot competite agains previously recorded training. My solution is to implement function to download trainings from the portal. It works like this: on the old device I upload my trainings to the portal, using already implemented method. After moving to new device I download specific training to the training log, with new desired funcion. This new function makes the RunGPS much nicer sofware then now.