Kimiri

When dates become green (reach what is known as the Kimiri stage) they are ready to eat, and are juicy and full of simple sugars like fructose and dextrose...

A text from ancient Mesopotamian city of Nuzi, lists green dates as one of fruits used as food in the city. Green dates are picked fresh and eaten locally, so they must have been grown in the area rather than imported... 

If this is so it means that the temperature at Nuzi must have been higher in the mid-second millennium BC than it is today. Dates cannot now grow in this region today as the climate is too cold...

Old Babylonian texts indicate that dates were harvested earlier than today. Although harvest times can vary from year to year and area to area, it would seem that the dates ripened earlier in Old Babylonian times than they do today, suggesting that the climate was warmer...

And we actually have (another) proof it was:

From "A STUDY OF DIET IN MESOPOTAMIA (c.3000 - 600 BC) AND ASSOCIATED AGRICULTURAL TECHNIQUES AND METHODS OF FOOD PREPARATION" by Elizabeth Rosemary Ellison, Institute of Archaeology, 1978