Parked at the Upper Canyonback Trailhead at the east side of the Mulholland Drive route and started around 7:30am on the 4th of July with 2 20oz handheld bottles so I didn't need to stop anywhere. I did a little out and back on the Upper Canyonback trail before hitting Mulholland Drive. It was already getting warm by 8am but the fire road was still pretty busy so I wore a mask when I had to. The route is all run-able on wide fire road until the last mile which goes between road, other pavement and dirt. There are a few intersections without signs so it is helpful to check out the route before hand on Strava. Turned around at Topanga Canyon Boulevard (Route 27) and headed back the way I came to finish the rest of my run. As I got closer to the East terminus, the crowds were growing so I recommend getting an early start to not have to dodge bikes and hikers. My two bottles were enough to get me back to my car well hydrated.
The route is all exposed so it can get pretty hot. There is water at the San Vicente Mountain Park about 1 mile from the East terminus but nothing obvious close to the west end of the route.