Page 26 troubleshoot: Pilot outage lists several things.
‘Dirty or clogged air intake screen’ causing less combustion air … this situation might be worsened by higher temperature which could reduce vent draft. Do the vent test because it was offered as specific solution to problem that was described.
Try getting more air to burner and pilot. Don’t install water heater in dusty environment.
Clean the air intake screen, and flame arrestor located inside the combustion chamber. Page 20 in manual
Remove burner and clean burner parts as shown in manual. Page 20
Clean flame arrestor
Are you running more fans during hot temperatures that might cause air to flow down the vent pipe and blow out pilot? Do the vent test as described to eliminate this possibility.
Page 26 troubleshoot also lists high ambient room temperature.
If attic or room reaches high enough temperature, the oxygen molecules separate far enough to cause oxygen depleted air that can put out pilot. Again the solution is to get more air to the combustion chamber by adding attic vents and cleaning air intake. Check for flame color and sooting as evidence of poor air quality.
Temperature might also affect the thermocouple connection. If thermocouple is too tight, it will short against the gas control and pilot goes out. So it is possible that higher temperature would cause slight expansion of metal that causes this short. Might want to go through checklist offered in original post to eliminate this.
The problem could be 2 or 3 things at once. Including bad gas control valve.