Rule of Thumb: There is no rule of thumb. :-)

To maintain the lawn, in most areas natural rainfall will be fine. If the lawn goes dormant during summer, it'll pop back in late summer to very early fall with no issues. I'm in that condition this year--the weather simply isn't very good for lawns, with higher winds and low rainfall, and I dislike throwing potable water at the lawn.

If you wish to keep it green regardless of conditions, it depends on the temperature (and winds). At lower temperatures, an inch of water every two weeks should be sufficient. At normal summer temps, an inch a week, delivered all at once, is the norm--but it will vary by soil type.

When very hot or hot and windy, half to three quarters of an inch every 3 to 5 days might be best, soil type depending once again.

The most accurate system available is to employ evapotranspiration data if you have a reporting station near you. Be forewarned, if you use that data as a basis for soil water replenishment, you'll likely end up a pauper. The best method is to follow the advice above- deep and infrequent.It works relatively economically.

Personally, I will not allow the lawn go more than a couple of weeks without at least 1/2" as I withheld water all of last summe and lost more than a third of my lawn. with overseeding and natural spreading it recovered by this last May. but IMO, not worth the savings in water.

This is not a rule of thumb,. but it seems to work as a starting point.

Watering: Deep and infrequent is the mantra for watering. This is for all turf grass all over the place. Deep means 1 inch all at one time. Put some cat food or tuna cans around the yard, and time how long it takes your sprinkler(s) to fill all the cans. Memorize that time. That will be the time you water from now on. My hose, sprinkler and water pressure takes 8 full hours to fill the cans. Your time will likely be less. I like gentle watering. As for watering frequency, that depends on the daytime air temperature. With temps in the 90s, deep water once per week. With temps in the 80s, deep water once every 2 weeks. With temps in the 70s, deep water once every 3 weeks. With temps below 70, deep water once a month. Note that you have to keep up with quickly changing temps in the spring and fall. This deep and infrequent schedule works in Phoenix and in Vermont, so it should work for you. The reason for deep and infrequent is to grow deeper, more drought resistant roots and to allow the soil to dry completely at the surface for several days before watering again. If it rains, reset your calendar to account for the rainfall.