Nope, not related to ambient temperature. The car can be baking in the sun the whole day, but as long as the fluid is cool, it will engage perfectly.

If the problem is as you say a "sticking apply piston or a warped steel plate", is there any other up/down shift that uses the same component as the P > D? Maybe I can look for sympoms there?

One issue the transmission has is sporadic harshness in some shifts, I believe caused by sprag issues. My heavy right foot takes credit for that

Could a failing/failed sprag cause the P > D harshness? If so, why only when hot?