Check your power options. By default laptops are supposed to be set to never hybrid sleep and to hibernate instead.
The desktop default of hybrid sleep keeps running the RAM at low power and also writes the hibernate file to disk in case of power loss. On a laptop such a setting would cause a constant drain on the battery when unplugged.
Fast startup shouldn't affect this as it is merely hibernate after logging you out.
The other thing I can think of is any BIOS setting that as mentioned keeps the USB ports powered or constantly watches the NIC or modem for wake-on-LAN
As the battery is removable (it's under a panel on the Envy 14) you can check if it rapidly self-discharges even when not installed, which would prove it is...