i would say fairly easily actually. running water is generally cooler than surface temperatures, and stores energy. whether heat, or cool. sitting on a cool river stone on a sunny day is all it would take to spark that idea. 💡
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Ok but no one else was running water through their city like that! One thing to have the idea, another to say yep not only will we run the water, we'll do it close enough to walls so that they're cooler
Not just Romans, many ancient civilisations used water and underground chambers for refrigeration. It's amazing how much we've forgotten with the ease of use of electricity.

Yakhchāl - Wikipedia
Ancient Origins Reconstructing the story of humanity
Ancient Advanced Technology: 2,400-Year-Old Yakhchals Kept Ice in the Desert | Ancient Origins
The yakhchal was a type of ancient ice house (refrigerator) built in the deserts of Persia - demonstrating the human ability to find solutions to p...