In order to support the new temperature badge feature, Perfect Weather had to register as an app that uses GPS in the background (and it does at times when you have the temperature badge option turned on). Unfortunately, this changes the way iOS treats the app, even when the temperature badge icon is not enabled.
I could bore you with the technical details, but suffice it to say, Perfect Weather is not actually using GPS in the background and draining your battery. The app only uses the GPS while active and immediately shuts off the GPS when you close the app. If you wait, the Locations Services badge will actually turn off on its own in 10-20 minutes.
For those who are skeptical about this explanation, here's a screenshot of an energy use profile of the app: http://assets.contrast.co/energy-use.png The app was launched and used for a few seconds, then left closed for 20 minutes. As you can see, there's a red line indicating GPS usage while the app was open, then as soon as the app is closed, the GPS is powered down and Perfect Weather isn't doing anything at all in the background (the CPU usage shown is just standard idle usage for iOS).
If you aren't using the Show Temperature on Icon feature, you can disable Background App Refresh in the Settings app to make the Location Services badge go away within 10-20 seconds of Perfect Weather being closed.