Where would you go to cool down if you were at a foreign city without money to spend in fancy venues, in a totally burning hot summer day?

Splashing her feet in a fountain…....sitting in the cool dark of a cathedral…..

Many stores are air-conditioned. Time to go shopping/browsing!

Museums!

public library

Church! (we were in Italy in summer and found that churces were quite cool!)

@mcbealer beat me to it. Library, definitely.

Cathouse.

beaches (don’t forget to put on a generous amount of sunblock with a high SPF)
malling

I would go to a park and lie under a tree

She could go see a movie. They tend to be climate-controlled (I know I saw a few French films to get out of the weather.) It also presents the culture in another way (as long as she doesn’t just see a dubbed film from here.)

i would go hang out in a hotel or resort lobby for a while, and just chill. they’re public places, and they don’t mind if you come in, sit, read, whatever.

also, if you’re ever anywhere and need to use the bathroom, go to a hotel and use the one in the lobby. they’re usuallly in pretty good shape, especially compared to bathrooms in other public places like fast food places or stores. nobody will question why you’re there, believe me, people go to fancy hotel lobbies just to view the architecture.

if you’re in a mall and want to wash your hands, go to bath and body works. they usually have a sink in the store for people to try out their soaps, so of course, they have a selection of their soaps there for you to try, and paper towels. (that was kind of off topic, but just a tip i thought of).