@Reel To Reel Is Correct, GPS Off No Prob. After Market Batteries And Or ORIGINAL Panasonic Batteires From ZS/8/9/10 Also True,.....However IF You Upgrade Your Firmware From 1.00To 1.10 The After Market Batteries Will NOT Work...I Have Many From Ebay .....Make Sure You Ask Before You Buy An After Market Battery And Or If You Upgrade Your Firmware......That It WILL Work With Firmware Ver 1.10 ......From Panasonic's Site....
DMC-ZS20 Firmware Update Version 1.10 Version: 1.10 Compatible Models: DMC-ZS20S , DMC-ZS20R , DMC-ZS20K , DMC-ZS20W Issues Addressed: 1.Improved picture quality for Handheld Night Shot MODE. Download Installation Instructions
GPS mode has two choices for turning it off. You can set it (1) to be off always (never on), or (2) set it to be on only when the camera is powered on, and turned OFF when the power is turned off. Otherwise, as you've noticed, it will drain the battery as it continues to perform GPS even when the camera is powered off.
I keep my GPS off to save battery usse. I have three batteries I use, 1 being the panasonic and the other two are after-market batteries. Cannot see any difference but then I have not did a firmware upgrade so I cannot speak to what happens then. I only got the ZS-20 so I could have GPS when I travel or out fishing in a boat and want to mark a good fishing spot, etc. Otherwise I keep it off.