If you put down seed now, and since it takes most good grass seed 4 to 6 weeks to germinate and start to grow that puts you into the middle of July when the weather is hottest and dryest and is not a very good time to be trying to grow most grasses. That is also the time of year when the cool season grasses will be wanting to go dormant because of that hot, dry weather.
The best time to seed cool season grasses, Kentucky Bluegrass, Perennial Rye, most fescues, is late August to mid September so when the seeds germinate they have the (normal) fall rains and cooler fall whether to get established.