"Hace mucho frío en enero."
Translation:It's very cold in January.
No, in these idiomatic weather expressions they're using a noun after "hace", so the adjective "mucho" is used.
Hace (mucho) viento (the noun "wind"). It's (very) windy. Hace (mucho) sol (the noun "sun"). It's (very) sunny.
More on weather expressions at studyspanish.com (Grammar Unit Two).