Utter bollocks but still plausible
First, all modern humans come from a single tight lineage no more than 100k-150k years old. Any skulls that predate the exodus of homo sapiens sapiens from Africa are irrelevant.
Second, the driver for human brain size is the need for social skills; humans being above all successful because they can construct large and complex societies, through pure brain power. It's a safe assumption that skull size follows brain size, with thickness of the skull correlating to the level of violence in the society, and overall skull size limited by the technical considerations of the human birth canal.
Lastly, the driver for human expansion out of Africa probably *was* climate change, so in a very indirect way climate change could be seen as the cause of larger skulls, but it's really very stupid of the researchers to confuse correlation with causation.
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