I'm with you. I've been here for 38 years and still can't get over the heat!
Don't know of any indoor play places here in NW Tucson, except for the playroom at McDonald's.
Try to come up with as many interesting indoor activities and games as you possibly can. A good website to check out is: www.familyfun.go.com, from the magazine of the same name. You'll find lots of crafts/ideas/recipes to do, many requiring items you already have on hand.
If your little one insists on outdoor play, here are some suggestions that may help:
Avoid swimming during the peak hours regarded as most harmful to skin and eyes. I think it's 10AM to 2PM, but not absolutely sure. Visors, floppy hats, or sunglasses are a must - as the sun's rays are responsible for premature occurance of cataracts and other vision problems. Studies have proven that just one "bad" sunburn during childhood can cause skin cancer decades later in life.
Keep plenty of hydrating drinks on hand and offer them often. My Pediatrician recommends Propel, made by Gatorade, which comes in lots of fun flavors, and contains a mere fraction of the sugar content found in fruit juice and other drinks.
Create an outdoor play area with as much shade as possible - if no covered porch, try rigging up some inexpensive shade cloth material available at Home Depot type stores.
A shallow pool in the shade is an alternative to swimming out in the sun. The smaller the better, as it's best to empty and wipe clean daily.
Misting systems are also good. I found a very cool little portable one at Ace two years ago. I think it was like $9.99. It attaches to a garden hose, and stands on it's own, shaped sort of like a snake, with misters along the vertical length.
Make sure you have mosquito repellent, as the little buggers will be out (especially at dusk), in full force during this next month or so of monsoon season. It takes just three days of standing water to hatch mosquitos. If bites and itches do occur, the best treatment I've found is Solarcaine - a bottle of green gel containing pain and itch relievers in an aloe base.
By the way, not sure where you moved from, but if you're unfamiliar with our yearly monsoon weather, please be aware that lightning can and does strike often, usually along the furthest edges (and beyond) of the visible storm/rain cloud area. If you can hear the thunder, it's time to get inside.
Even in the shade, the harmful rays of the sun can reflect off clouds, walls and other surfaces and cause damage. Find a sunscreen with high SPF - 40 to 50, and water resistant properties and slather it on as recommended. A quick soapy rinse off once indoors will easily remove bug repellent and sunscreen products.
Oh, and one more thing - I work outdoors every day in this heat, driving a vehicle with no air conditioning, and wearing a required uniform consisting of 100% polyester shorts and 65% poly shirt. I can't tell you how disgustingly uncomfortable this is, but I will tell you that the rest of my wardrobe consists entirely of items that are 100% cotton with no exceptions! I've found that even those "cotton" shirts or jeans with 3% lycra or poly for "stretch" are absolutely intolerable in temperatures above 80 degrees. So... stay as cool as possible with lighweight all-cotton garments.
I apologize for my lengthy response, but I can't help it. It's just my way of offering as much help and info as I can! Good luck to you and your little one. I'll be praying, as always, for November's arrival!
L. B.