Do Birds Get Cold Feet?

I have been enjoying watching the birds drinking and taking baths in the birdbath outside my kitchen window. It requires being filled twice a day since the water gets dirty from it's many bathers of all different sizes and varieties. In this extreme heat and high humidity, even the birds must feel it since so many of them are opening their beaks as if to breathe through their mouths. Well, my hubby decided they deserve a treat, so he filled the birdbath with ice cubes. I have yet to see any reaction, so the verdict is still out on this one. I best get back to watching again before the ice melts. I hope this doesn't cause any birds to fly South prematurely.