0.4. The highest average temperature in Big Sky is July at 71 degrees. The coldest average temperature in Big Sky is December at 20.3 degrees. The most monthly precipitation in Big Sky occurs in June with 2.6 inches. The Big Sky weather information is based on the average of the previous 3-7 years of data.he Big Sky weather information is based on the average of the previous 3-7 years of data.