My boss drives a Ford E150 (96 or so, V8). I often drive the van into town for lunch, but I am unable to shut it off now that the weather is hot, because the van will not startup again. This happened during a warm spell a month or so ago. Given anywhere from an hour to two hours, the van will start up again. It's about 80 degrees F where I live.

Turning the key results in no clicking sound, other than the normal beeps, but the light indicators and whatnot seem to turn on.

It's a weird problem. The van will start up if you shut it off and immediately turn it on again. Also, if it is cool, you can start/stop it with no problem. The temperature gauge is at about midway when shut off, and once it gets pretty low the car seems to start again. Installing a new starter did not solve the problem.

Any idea what the problem could be?
Thanks for any help.