Froid vs. froideur
The French words froid and froideur both relate to low temperatures or a lack of warmth, but they serve different grammatical roles and contexts. Froid is a very common adjective and noun used primarily for literal, physical temperature and the sensation of being cold. In contrast, froideur is strictly a noun meaning coldness, and while it can describe physical chill, it is predominantly used in a figurative sense to express emotional distance, indifference, or a lack of affection.
Froid
The word froid acts as an adjective meaning cold, or as a noun meaning the cold. It is the standard word used to describe low temperatures in the weather, the physical sensation of being chilled, cold objects, and occasionally a distant personality.
Il fait très froid au Canada pendant l'hiver.
(It is very cold in Canada during winter.)
J'ai oublié mes gants et maintenant j'ai froid aux mains.
(I forgot my gloves and now my hands are cold.)
Ce plat est délicieux même quand il est mangé froid.
(This dish is delicious even when eaten cold.)
Il a attrapé un coup de froid en sortant sans manteau.
(He caught a chill by going out without a coat.)
Son regard était dur et froid.
(His gaze was hard and cold.)
Froideur
The word froideur is a noun that translates to coldness. It emphasizes the state or quality of being cold. It is most frequently used metaphorically to describe a lack of human warmth, emotional detachment, or hostility in relationships, though it can occasionally refer to the physical chill of an object or environment.
La froideur de sa réponse m'a beaucoup blessé.
(The coldness of his answer hurt me a lot.)
Les nouveaux voisins nous ont traités avec une grande froideur.
(The new neighbors treated us with great coldness.)
Malgré la froideur de la pièce, il a réussi à s'endormir.
(Despite the coldness of the room, he managed to fall asleep.)
On peut sentir une certaine froideur entre ces deux collègues.
(You can feel a certain coldness between these two colleagues.)
La beauté de ce bâtiment est gâchée par la froideur de son architecture.
(The beauty of this building is ruined by the coldness of its architecture.)
Summary
In short, choose froid when you need an adjective to say something is physically cold, or a noun to talk about the literal cold weather or bodily sensation. Choose froideur when you need a noun to express the abstract concept of coldness, particularly when describing someone's emotional distance, a lack of affection, or an unwelcoming atmosphere.