During the previous track day at Vålerbanen in July 2020, I had to run the engine at maximum of 5500-6000 rpm to keep the oil temperature below 140 °C if the session lasted more than 10 minutes. The C5 Z06 does not come from the factory with an oil cooler, but it is clearly required to avoid overheating the engine.
The stock temperature sensor is M12x1.5 while the oil cooler block is 1/8 NPT. I decided to drill and tap the correct M12x1.5 thread, which worked out nicely. I thoroughly cleaned out debris using brake cleaner and compressed air. 
Oil cooler brackets being made
Finished brackets
New oil cooler block installed. The part of the hoses passing by the exhaust manifold are protected by additional heat insulation. This block has no thermostat, as I simply cover the oil cooler with a sheet metal plate when not doing track days. A thermostat also restricts the oil flow, by the nature of its design.
Oil cooler installed. I decided it had to be installed in front of the radiators to maximize the efficiency. Notice that I haven't used any 90 degree fittings, as they can cause flow restrictions. It might be that the hoses are bent too tight, so I'm considering adding full flow 90 degree fittings later.