The last couple of times you cut the lawn this year, do so with the mower on its lowest blade-height setting. Cutting so closely allows more sunlight to reach the crown of the grass blades, and it may minimize the extent to which your lawn looks brown while dormant. Just be careful not to trim off any more than a third in a single session. Overly aggressive cutting can shock the lawn and compromise its health.
Related: The Right Way to Mow the Lawn