It's quite easy to repro - install GCM and switch on Bluetooth, battery dies in roughly 12 hours. Switch off Bluetooth, or uninstall GCM and the phone lasts 3-4 days.
This is a Moto G without a SIM, so airplane mode with WiFi and BT usually left on. I'll try and raise a case with Garmin, but they may well just point out that the Moto G isn't on their (tiny) list of supported Android handsets unfortunately...
It definitely didn't used to be like this, but can't remember which recent GCM update introduced it. There seem to have been so many in the last couple of weeks. Can anyone remember which update made it noticeable?
I resume this thread to report that since the last update (3.3.2) I have the problem of increased battery consumption. I'm sure that id depends from Garmin Connect App because I solved it by disabling notifications to the app from system settings.
I believe in this way the service that listens for connected devices is no more active, I also believe that service is the responsible.
Can You explain how did you disable the notification service?