I'm new, but I think the heat could kill the worms and it will attact ants and flies. I would suspect there could be a number of problems with that idea.

Warmth
Red Worms (Eisenia fetida) tolerate bed temperatures as high as 43C (109.4F). It is best to avoid letting your worm bed temperatures go above 30C (86F). Adults of this species have been known to survive being encased in frozen material, and the cocoons are well adapted to survive sub-zero temps for extended periods. The optimal temperature range for breeding Eisenia fetida (red wigglers) is 15-20C (59-68F), yet maximum growth (weight gain) occurs closer to 25C (77F), if you want to go fishing.
I am in Winston and worms do OK in the garage here.

Does your garage get pretty warm? We have a storage shed but I'm wondering if it would just get hotter in there than it would in the open air.

We just went through a hot summer, and I simply removed the lids, put a layer of damp hessian on top and hept it damp. The evaporation helped keep the bins cool. They were in the shade of course.

What Do Worms Need?
Moisture
Earthworms breath through their skin and thus need to stay moist at all times. Many people refer to the ideal moisture content of worm bedding as being similar to that of a "wrung out sponge". This is an easy determination of moisture content that has been accepted for 30 years or so. Newer research has shown that composting worms typically prefer higher moisture content.
Oxygen
Even with much lower oxygen requirements than humans, composting worms still need to breath. Store your self-harvesting vermiculture system under some type of cover out of rain (too much water) and direct sunshine (too much heat). A garage, utility room or the basement may be some of the best places to put your worms. The composting action in the system creates heat which pulls air in the bottom and pushes it out the top, just like a chimney. The activity of the worms themselves will also aid in aerating the system, so unlike a hot composting pile, you wonÂt ever need to Âturn your self-harvesting vermiculture system contents.
Warmth
Red Worms (Eisenia fetida) tolerate bed temperatures as high as 43C (109.4F). It is best to avoid letting your worm bed temperatures go above 30C (86F). Adults of this species have been known to survive being encased in frozen material, and the cocoons are well adapted to survive sub-zero temps for extended periods. The optimal temperature range for breeding Eisenia fetida (red wigglers) is 15-20C (59-68F), yet maximum growth (weight gain) occurs closer to 25C (77F), if you want to go fishing.
Food
It is actually the microorganisms growing on waste materials that provides the main source of nutrition, not the material itself  but the worms do manage to eat the rotting material as well. The best food sources support the richest microbial populations. This helps to explain why animal manures (NEVER DOG or CAT FECES) are the best food for Red Wigglers. I recommend setting up your self-harvesting vermiculture system (with bedding and ÂfoodÂ) before you get your worms. Then there will be a very healthy microbial community waiting to feed the worms you add later. If the worms are "overfed" (Too much raw food and not enough microbes), the worms can get sick and die. To give the microbes a "head start" in the composting race, mix a little finished vermicompost in with the worm food.
Darkness
Worms are protected from light by covering the feeding with shredded cardboard each time. This will also help to stop gnats and flies from laying their eggs in your self-harvesting vermiculture system.
Wurmz iz e-z!
Garage=Shade=Darkness
Moisture=70-80% Water=Worms Breathing
Food=Brown(Carbon)X3&Green(Nitrogen)X1=Microbes
Oxygen=Aerobic Habitat=Healthy Wurmz!

The cockroaches would freak me out...shudder...An in-ground pit is the solution for both hot and cold weather. But lacking an actual in-ground pit, I suppose that you could sink a plastic bin into the ground and cover with a wooden A-frame to shade, and keep it cool and moist.

"The composting action in the system creates heat which pulls air in the bottom and pushes it out the top, just like a chimney." That is exactly what I would like to belive. No scientific evidence I know of supports it. But still I want to belive. Until someone puts air sensors or something at different parts of the system I will raise one eyebrow. How does air get through a solid?

"cockroaches" I have so been waiting for someone, somewhere to post about this in relation to bins. This is the first I have ever seen of it on line. It is reported as a non problem. I'm glad. If you have invented a bin "they can't get into" you should patent it. If I saw any bug I did not want the bin would be outside not near the house. I would think a bin would be an ideal hangout for unwanted friends. If I was in unwanted friends mode I would dump everything heavy overboard. I understand in hot areas of the country the battle might be not able to be won. Every area has their special vermi issues to learn how to deal with. The internet is a great place to share informaiton.

"Moisture, Oxygen, Warmth, Food, Darkness: Chapter headings if I have ever seen any. 15 years is a long time to go between informative books on a subject. A lot of composting has happend since then and a lot of composting knowledge has been gained.

I think we(this forum and others), are the new book on vermicomposting. This book just needs an editor.