In the following sentence:
Les poissons sont des animaux à sang froid.
I don't understand why the proposition à is used. The meaning is likely animals with cold blood, but then why not use avec? I learned that some food can use à with ingredients to mean something with extra ingredients. For example, café au lait. But it seems to me that des animaux à sang froid does not fit in this usage.
So why does it use à as a proposition?
You have encountered one example of the so-called complément du nom (in brief CDN). This is a word or a locution that specifies, or completes, the noun. It is right after the noun it specifies and therefore belongs to a nominal group. E.g.
(Vous avez rencontré un exemple du soi-disant complément du nom (en bref CDN). C'est un mot ou une locution qui spécifie ou complète le nom. Il se trouve juste après le nom spécifié et appartient donc à un groupe nominal.)
La robe de mariée. Le verre à pied. La valise en carton.
The CDN is usually after the noun and it is often (but not always!) introduced by a preposition: à, de, par, pour, devant, derrière, sans and so on. Examples of the former usage include:
(Le CDN est généralement après le nom et il est souvent (mais pas toujours!) Introduit par une préposition: à, de, par, pour, devant, derrière, sans et ainsi de suite. Voici des exemples de la première utilisation:)
Le sac de Gabriel. La voiture devant le garage. Une chambre sans lit. Une chaîne en argent. Une machine à pression. Des animaux à sang froid.
Examples without preposition: (exemples sans préposition :)
La Tour Eiffel, projet recherche, responsable qualité, débat marathon, la stratégie Mitterrand, les années Poutine, etc. (see the questions in FSE: Juxtaposition des noms (substantif plus substantif) and Complément du nom sans préposition ?)
See the links below for more details:
Regarding your initial feeling that "[t]he meaning [of animaux à sang froid] is likely 'animals with cold blood ...,'" it's interesting (and perhaps even relevant) to note that avec is at/near the top of CISCO's (via CNRTL) list of synonyms for à (seemingly tied with de, which similarly has avec listed as its top synonym).
The fact that avec and à are/can be synonyms indicates to me that the gist of your initial feeling is more accurate than you imagine, stopping just short of being totally accurate for the simple (and understandable) want of being aware of all the particular cases where
"À signifie « avec », « qui a »" (À signifies "with" or "that/which has"),
including the following case found in the entry for "À" from TLFi via CNTRL (which to me, granted a non-native speaker, would seem to include/cover the term in question):
D.− Le substantif déterminé par le complément introduit par à est un substantif de sens concret (désignant des êtres inanimés ou des animaux)
...
- À introduit un compl[ément] d'accompagnement.− À signifie « avec », « qui a » :un char à banc « avec bancs / qui a des bancs » :
a) À + subst. non actualisé :
un arbre à feuilles conifères ... les bêtes à cornes ... un bonnet à poil ... la bourgeoisie à talent ... un chapeau à fleurs ... le char à banc ... ... une femme à cheveux bruns ... un habit à queue ... un homme à femmes ... un homme à moustaches ... un homme à talents ... une moquette à fleurs ... des rentiers à lorgnons ... une robe à carreaux ... un serpent à lunettes ... des souliers à talons ...
The reason why not to use « avec » is the same in French as it is in English. Why say "cold blooded animals" in English instead of "animals with cold blood? Notice that "with" is rare; If I ask you why is "with" rare, you are bound to say something like "because that's the result of a taste in matters of language that is particular to the English people." and considered as a global assessment that's what it is. In this particular case, of course, there is the evident enough, partially explanatory reason of the propensity of English for compound nouns without preposition. But then, you can ask what has brought this propensity about, why didn't the English see in the use of a preposition an improvement in the construction? First of all, is there an improvement? If you want to reach at deeper reasons, the why of a particular taste, the understanding of what has influenced a particular choice you often end up into a quagmire of possibilities or the dead ends of no more recorded information. In any case you're most likely bound for delving into the abysses of linguistic history in time consuming research. There are myriads of questions such as that one to which the most relevant explanation for the non-specialist is "It is so because that's the way the turn of things made it in our language.".
For instance, "animaux de sang froid" has not been retained in French but it is an equally valid possibility; to wit the constructions "un homme de constitution robuste" (equally used and synonymous, un homme à la constitution robuste), "un animal de férocité peu commune", "gens de grande taille", "personne de grande taille"; someone's height is one of his/her attributes just as the particular type of blood of a creature is one of its attributes; why "à" in the one case and "de" in the other? Why "un homme de petite taille" and "un homme au teint basané" and "un chien à poil ras/long" ou "un enfant de sang mêlé"? There isn't much else to justify that than a general reference to usage, usage has made it so. There is of course a certain logic in the meaning of prepositions, but that is far from perfect and certain prepositions have no real meaning; I'll take the case of the preposition used in the constructions of comparisons in English for the word "different"; there isn't just one word but three: "from", "to", "than"; they are more or less markers, not conveying a definite meaning. The state of affairs highlighted by this example is something to take into account in this question of why one and why not another, and that is that there exists a given degree of incertitude as to the meaning of the very words we choose and define for the needed functions.
