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I'm guessing that falling rain equilibrates fairly quickly to the temperature of the air it's falling through. However, it isn't always at the exact same temperature, which is why we can get the lovely phenomena called "freezing rain".
How far below the ground-level air temperature can rain be?
Rain is slightly cooler than near-surface temperatures because it condenses at higher altitudes where the air is colder. Thus rain tends to reduce the temperature difference between the surface and higher altitudes. This temperature difference is the free-energy or exergy source that ultimately drives convective weather systems and storms, and is expended or diminished by their operation. Usually the temperature of rain is only a few degrees cooler than that of ambient air by the time it falls to the surface. An extreme example is hail --- because of the large size and rapid fall of hailstones they can reach the surface still frozen even in midsummer. By contrast, freezing rain is often a different phenomenon. Freezing rain often results when rain originating in a warm air mass aloft falls through a subfreezing air mass near the surface, as typically occurs ahead of an advancing warm front.
