Starter

I have a strange issue. I just got a brand new battery and had the cables checked as well. When I drive in town for ten to fifteen minutes at a time, I have no problem. But when I drive for thirty to forty minutes and I turn off the car, it will not start at all. No sound, no nothing. I mean when I put the key in the ignition, boom, all the bells and sounds, bright light etc, If the car sits for three to four hours and completely cold engine. Boom starts up and I am on my way. But, as soon as I try it again after thirty five minutes of driving same thing again. Thoughts anyone? So appreciated.