Private land operated by the Nature Conservancy, Davis Mountains Preserve. Check website for open dates/weekends when it is free to the public.
Bridge Gap Road leads to the short Livermore Summit Trail and the summit of Mount Livermore. At about 1.5 miles in, the trail diverges with Limpia Chute Trail to the left. Limpia Chute is easier and a nicer hike through tall trees and great views. Bridge Gap Road during that portion is very loose and steep. The road eventually ends at the short Livermore Summit Trail which will require a scramble to get to the top of Mount Livermore - the last ~200' needs to be done with caution. There are great views at the top.