Cold weather grey water cold frame

I will be putting grey water into mulch basins sized according to Art Ludwig's book. Would putting an (empty) cold frame over them help to keep them infiltrating water during the winter? Here in Colorado the little snow we get in dry winters mostly sublimates off the surface, so recharging the mulch and soil during the winter would be helpful, especially for trees. (Winter watering of trees and perennials is recommended here, since trees often dehydrate when the soil is frozen and the days are warm and sunny.)

It would help, not sure how much but I would think an extra 10 degrees at least.

Have you already tried running greywater to mulch basins and they freeze over and run off? Because we have a cold winter here, but even so the warm greywater soaks in to some extent, and doesn't make toooo large of an ice field. In this dry climate, there are often sunny days in winter, and the ice seems to keep soaking in or sublimating even as new stuff arrives. And we don't even have mulch basins, we just run it right onto the bare garden soil.

No, I have not tried it yet. So potentially the cold frame would be unnecessary. That is good to know.