Regardless of the time of day you decide to fly, the ticket price will remain around the same. Book flights from Washington, D.C. Dulles Intl Airport to Gaborone as you normally would.
Due to multiple factors, expect to potentially pay more when booking in August. For low rates, October can be a great choice for travel to Gaborone. Flights to Gaborone from Washington, D.C. Dulles Intl Airport in October are about $1,379 on average during October, but can be found for as low as $950.