At minimum, it depends on how the power in your car works.
Some vehicles keep power on all the time. Some will only activate power when the vehicle turns on.
As for whether or not it will power on, you can do that test yourself. Take the device and unplug it. Does it turn off? Now, plug it in. Does it turn on? These are the same things that would happen in your car.
(I say to test yourself because I don't have the device to test it, but it's a simple test that you can perform yourself)
My car does turn the power off/on to the outlets when the car is started/stopped.
I don't currently have one, or I would test it out. I was hoping to find out the answer before purchasing it.
So- I purchased the device, and it does exactly what I wanted. As long as the battery is in, and the outlets in your car get power when the car is turned on... the hotspot immediately powers up and starts working. When I turn my car off, the hotspot continues to work until the battery runs out. When it's not being used, the battery lasts an amazingly long time.