"Hace mucho frío este mes."
Translation:It's very cold this month.
'Frío' is being used as a noun here, and therefore takes 'mucho'. (The sentence literally translates as, "It makes a lot of cold this month.")
If instead you were saying, "está muy frío", you'd be correct to use 'muy' before the adjective 'frío'. But you'd never use this expression with regard to the weather; it can only refer to a cold person or object, or to something abstract like a cold [uncaring] personality.