Does clarified butter harden when you put it in the refrigerator?

 Um-hum.

Clarified butter is essentially solid at room temperature. Putting it in the refrigerator is only going to make it solidify faster and more firmly.

When you clarify butter, you are removing water, milk solids, and salt (if it's salted butter). Removing the water – the second largest component of butter after the butterfat – is going to make it more likely to solidify at lower temperatures, not less.

If jars of ghee are available in the ethnic aisle(s) of your supermarket, you will see that they are solid butterfat at plain old supermarket temperatures. Because it is pure fat, it is not prone to spoiling, and so can be stored at room temperature for weeks.

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