Yes, in that sentence “chaud” is a noun that means “heat”, and as such its gender does not depend on who is feeling hot.
Notice that the verb is avoir (to have), not être (to be). We don’t say “I am hot”; literally, “j’ai chaud” means “I have heat”, but more accurately it means “I am feeling heat”. The same construction appears often: “j’ai froid” = “I am feeling the cold”, “j’ai faim” = “I am feeling hunger”, “j’ai soif” = “I am feeling thirst”, “j’ai envie de […]” = “I am feeling envy for” (= “I would like”) and so on.
Note that the more common translation of “heat” is “chaleur“, a feminine noun. The two are not interchangeable, though. “Chaud” as a noun is almost always used in the expression “avoir chaud” (to feel heat) or “être au chaud” (to be in a warm place). Other than in these two expressions, it’s safe to assume that you need to translate “heat” as “chaleur“.
Simply, no number or gender matching with “avoir” unless a direct object exists and comes before avoir, but this is a topic that is off this question.
Avoir chaud is a set expression where chaud is used, depending of the grammarians, as a noun (Robert), an adverb (wikipedia, cordial, Larousse, TLFi), or an adjective (adjectif adverbal: Gabriel Wyler, Martin Hummel) …
The noun chaud is masculine so can’t agree with anything. (J’ai grand chaud)
The adverb chaud is invariable.
The adjective chaud can agree in French, but there is no example of agreement with "avoir + adjective".
The nominal status of chaud can be challenged when we see it commonly used with an adverb like très:
Elle a très chaud.
Elle a très faim.
In any case, set expressions (locutions) should be considered as a whole and it can be pointless not to do it.
Here are some references where chaud is considered to be an adverb:
TLFi
A.− Emplois adv.
- Loc. verbales
…
c) Avoir chaud. Une rose, qui a trop chaud, se dévêt de ses feuilles, une à une (Renard, Journal, 1902, p. 762).
− Domaine de la personnalité hum. Autrefois son âme avait froid, maintenant elle avait chaud (Hugo, Les Misérables, t. 1, 1862, p. 506).
Larousse
chaud, adverbe
….
Avoir chaud, avoir une sensation de chaleur.
Cordial
chaud, adverbe
Dans les locutions "boire chaud", "tenir chaud", "manger chaud", "avoir chaud", "à chaud", "au chaud", etc.
Wikipedia
chaud, Adverbe Invariable
…
Se tenir chaud. Boire chaud. Manger chaud.
Dérivés
avoir chaud
…
The agreement wouldn’t have been wrong with the verb être though, but beware of the different meanings:
Je suis chaude / Je suis chaud…
Avoir chaud, avoir faim, avoir soif, prendre congé, perdre patience, avoir peur, etc., sont des locutions verbales. Une locution verbale est une action représentée par un verbe additionné d’un autre mot. (cf. wiki) Chaud peut être à la fois nom et adjectif :
Ce plat est trop chaud (adjectif). Je crains le chaud (nom).
Les locutions verbales font partie intégrante du verbe en lui donnant son sens; les deux mots “avoir” et “chaud” exprimant une seule idée.
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