Toyota Repair: Toyota Corolla, frozen veggies, toyota corolla

Question
Here's how it goes. I drive my car - say to the store, or to work. I turn it off and go inside, come back out anywhere from a half hour to two hours later and the car won't start. No turn over AT ALL. It's NOT the battery. Everything turns on - fans, lights etc. It's just that Last click of the key - it has no response in the car.

Mechanics are all baffeled by this (I've spent over a thousand dollars on it now). None of them can get it to happen so they can't properly diagnose it.  

It's NOT the starter - I had it replaced.
It's NOT the ignition. Had that replaced too. I also recently got a tune up.  I take very good care of it, lots of oil changes etc.

Here's what I think now - it's some kind of overheating problem. Like the computer in the car think's it's too hot and won't let me tune it on. Usually when it won't start I open the hood and fan the engine for anywhere from ten minutes to an hour and then it will start. Today - I even  (this is embarassing but I had an appointment to get to) ran in the house and got a bunch of bags of frozen veggies and laid them all over the engine - it started five minutes later.

Iremember a Honda Civic I had once that had a fan that would tunr on when I got out of the car.  I never hear a fan turn on in this car - SHOULD IT?

There is nothing else wrong with this car - I love it. Here's another thing - it NEVER happens between October and April. It only happens in the summer months.

Good Luck!  

Answer
Very unusual problem, I need to ask a few questions and maybe after that some more, when trying to start and it doesn't, does the engine not turn over at all or does it sound like it is turning over but not start up, this is important. I think you mean that it does nothing but I need to be sure. Is there any noise at all when trying to start it? The computer only has control of the engine once it cranks over, it does nothing until it sees the engine turning over this is why it's very important that I understand whether the engine is turning over or not. By the way what year is the corolla and does it have an automatic transmission or manual. Does it overheat according to the temperature gauge on the dash? What did you do with the vegetables after it started? Just kidding.  Is there a check engine light on and did it come on in the past? The problem is obviously temperature related but in most cases, depending on what year it is the check engine light will turn on if there is a problem with any of the engine control systems, the more information you can reply with the more I can pick my small brain to have some kind of answer for you, please reply directly to me at tedrttr@yahoo.com>