Why don't you just try it yourself for a couple of days and see how it affects your battery? Do you make it to the end of the day on a charge with it on? If not, turn it off and check the next day.
Everyone gets different battery life. If you live too far away from a cell tower, attention aware is the least of your problems. If you constantly watch videos, bye bye battery.
Are you actually having battery issues or are you just looking for things to be nervous about? That seems to be everyone's favorite hobby here in Negative Nancyville.
FYI, attention aware is designed to, among other things, dim the screen when you're not looking at the phone. So who's to say you might actually lose some battery life by turning it off?