With Rachio watering correctly for me on my grass (every 2-3 days) in this heat, mushrooms are no problem. I had mushrooms a lot before Rachio because I would over water. You get mushrooms when the top of your soil stays wet, so if you are watering every day, or making the mistake of watering twice a day due to heat, then you'll be at risk. But if you are deep watering every few days, the mushrooms can't establish because the top of the soil is drying out. Also the timing of your watering, make sure you water in the early morning and not nighttime.
Now I will add that for us in AZ in monsoon season, I did have some issues with mushrooms because the Rachio didn't fully compensate for the soil and weather conditions that hit. I will try to keep ahead of that issue in August.