The phrase “Dog Days” refers to the hottest days of summer. The Old Farmer’s Almanac lists the traditional timing of the Dog Days: the 40 days beginning July 3 and ending August 11, coinciding with the heliacal (at sunrise) rising of the Dog Star, Sirius.
The rising of Sirius does not actually affect the weather (some of our hottest and most humid days occur after August 11), but for the ancient Egyptians, Sirius appeared just before the season of the Nile’s flooding so they used the star as a “watchdog” for that event. Since its rising also coincided with a time of extreme heat, the connection with hot, sultry weather was made for all time.
More about the Dog Days can be found here: Dog Days of Summer