Yes, it does.
You have to tell it what you want to be done on law battery though.
You can configure all this in /etc/UPower/UPower.conf
Just read through it. It is self explanatory.
You can choose to measure battery tendency in percentage or time.
Those numbers will trigger warning, critical warning and action.
Final action can be PowerOff, Hibernate, HybridSleep
For obvious reasons there is no point in suspending on law battery.
P.S.
As a side note. It will not damage your battery on full discharge. It will make it last longer. You have to let it discharge completely at least once a week if you want your battery to last. So it is not the battery damage you should be worried about when your computer hits law. The filesystem damage and losing your files - that is your real problem.
So set your upower action to hibernate at 2%. In this case you discharge your battery and keep your filesystem safe. Do not set it for 1% as the power might run out just when it is in the process of writing your memory to swap partition.
Make sure the hibernation actually works manually. Run sudo systemctl hibernate. If it does not work configure the hibernation first which is a different story.
