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Hello I will help you with your question,
Can you try heating the computer located on the engine side of the firewall from the inside fuse block to see if that is the component being affected by the heating?
Not too hot just warm it up.
I am not sure what in the fuel pump circuit would be affected by heating under the dash fuse block as all the control for the pump is on the other side of the firewall.
Let me know what you find so we can work this out.
I don't know what else might be under there, do you have any aftermarket alarm that might be spliced into the pump wiring?
I don't know what would be affected by temperature then, all the fuel pump components are under the hood. The computer gets the signal the key is on and the engine is turning so it grounds the relay turning the pump on.
Sorry but without being there I don't think I can give you an answer so I will OPT OUT of your questions and maybe someone else will have a suggestion for you