I will check the exact time either to make sure I give enough time before trailing or to kind of ingrain in my mind how deer are moving through that particular area.I want to see if behavior repeats itself for future reference.
I always check the time right after I shoot a deer, so I can know when to start tracking it. I do remember one time. I shot my first deer with a bow at 8:06 AM :D
For sightings I will generally check once they're out of sight, and approximate. That information goes in my journal. My buck I shot this year, I sent a text almost immediately so I knew the time to the minute.