How to beat the hot August heat in Florence and Rome

wear light coloured cotton shirts, wear a hat , keep well hydrated, plan for some indoor activities (churches, museums etc) between 12 and 3

Museums holding paintings seem to be reliably air-conditioned. Archaeological museums, not necessarily. So schedule the Borghese in the early afternoon (if it's open then) and hit those churches!

To find places for lunch that are air-conditioned, look for signage that so indicates or at least a door that is closed. In my experience, an open door means they're praying for air-circulation to do the job, in which case it's usually cooler outdoors under the umbrellas, but that seems to be a smoking area.

On dreadful days an in-room picnic with food you've foraged is not a bad idea.

Although most small markets have bottles of cold water at good prices, you may not always see one when you're parched. Most bars sell bottled water (usually cold) for less than 2 euros as long as you drink it standing up.

There's a product someone recommended here last year that might be worth trying: Kool Tie.

Water from fountains is cool in Roma, you can refill your bottle from a specific spigot or a Nadine, a free-standing drinking fountain the size of a fire hydrant.

Cross the street to walk in the shade when necessary.

Churches are generally quite cool. I duck into a lot of churches as I walk past. I just don't wait for the big popular ones. I have discovered many of my favorites that way.

Churches are generally quite cool.
That's true, but requires meticulous planning. Most churches are closed for the best part of the afternoon (12.00 to 15.00 or 15.30).

Zoe fell victim to autocorrect. What she called Nadine are actually called "nasone" which means "big nose". Here's what they look like and how they work: https://formaggioheads.files.wordpress.com/2012/06/img_0368.jpg Block the hole and water will arc out from a hidden hole, allowing you to take a drink.

The water is delicious and 100% safe to drink and to fill up your water bottles.

As for popping into a church, it's true that many are closed during the early afternoon. When you do find one that's open, see if there's a lower level underneath the main sanctuary. That will be even cooler.

Pray for below average temperatures. Some years it happens.

You will be outdoors for most of the day, I don't think you are going to Italy to spend all day inside an air conditioned hotel. Churches are cooler than outside since they don't let too much sunlight and have high ceilings. Museums, such as the Uffizi, must be kept at a very specific constant temperature (about 18 degrees centigrade, or 65 F) for preserving the art pieces, although in some days, when it's very hot and there are lots of people inside, the a/c may have a hard time keeping it down at 18 degrees. But outside, the only relief is shaded areas and gelato.

Get outside early, while it's cooler. Dine late. More sightseeing is open all day now, but you can still consider an afternoon break, if you're worn down. I probably wouldn't be eating anything but gelato and cold beverages at mid-day.