So I get that socially in most places it is not acceptable for women to go totally topless. We can argue about double standards, but I'll let other people do that.
What I actually have noticed is that nowadays in a lot of Western cultures, women can get away with just wearing just a bra with no shirt in public. No, it's still not being totally topless, but it is a lot better heat-wise than wearing both a bra and then a shirt on top. Just it can't be like lingerie, but more like a sports bra or something. I've done it when heading into a shop after going for a run and not gotten any complains. I see others just occasionally walking around in something a sports bra when it's hot out, not even relating to intense exercise. Try it.