Does the Chevrolet Corvette C7 ZR1 have cooling issues when driven on a road course?

No, the C7 ZR1 does not suffer from the track cooling issues that plagued the earlier C7 Z06. Owners who track them report that the ZR1 has no cooling problems on a road course ¹.

Generally speaking for this platform, Chevrolet specifically engineered the ZR1 to fix the Z06's thermal limits. They completely redesigned the front fascia to feed a massive array of 13 heat exchangers, ensuring engine coolant, oil, and charge air temperatures stay in check even under sustained track abuse.

However, while you won't be fighting temperature gauges, you will be fighting brake wear. Because the ZR1 achieves such incredible speeds on long straights, slowing the car down from 165+ mph lap after lap takes a heavy toll on the brakes. You can expect to go through track pads and rotors very quickly ¹.