hald, I live in Vermont. We run a hose (3 actually, hooked together) up the hill to the garden. Most of it runs through heavy shade and yet when I turn the water on it runs not just "warm" but downright hot for several minutes. Hot enough to poach fish or eggs. Just from the water sitting in the hose in the sun.
I would suspect that much like houseplants, tomatoes grown in pots would like balance as much as anything - drastic change in temperature tends to shock them. That said, I would still acquire a thermometer and run a check on the temp of the water sitting in the bottom of that box.
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