Your work time will continue to record even if you device is no longer running. However, this could lead to inaccurate entries. Additionally, if you are asked to present copies of your logbook to an enforcement officer you will not be able to do so and could face fines and demerit points.
Just like you make sure your vehicle never runs out of fuel, you should ensure you can keep your phone charged when using Logmate. It's an essential work time, please treat it like one. The App will alert you when it detects your phone's battery is getting low.
Pro tip: Having a charger in your vehicle will make keeping your phone charged easy.