Speaking for the northern US, grass grows in the spring and again in the fall. I took an agriculture extension class when I had a lawn in eastern Michigan. The main points I took away were:
—Plant grass in April after the last frost or in September
—Don’t worry about brown grass in the summer – your lawn doesn’t require golf-course levels of watering – it can go dormant and will get green and lush again in the fall
I would check your nearest extension for info specific to your area, but for starters here are a couple of articles from Michigan State.
Tips for seeding lawns in September – September is the ideal time for seeding turfgrass in Michigan
Summer Lawn Care