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Above the weather there is a layer where the air gets warmer the higher you go, not cooler.

Earth Science Facts, sourced to UCAR Center for Science Education.

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In the layer we live in, temperature falls with height, so warm air rises and keeps rising and that is what builds storms. The stratosphere reverses it, because ozone up there absorbs ultraviolet sunlight and heats the air from the top down. Warm air sitting on cold air is stable and refuses to churn, so rising storm clouds hit that boundary and spread sideways into the anvil shape. Airliners cruise just inside the calm.

The claim comes from UCAR Center for Science Education.

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