Hoping someone else would explain it in detail, let me just give you a tiny bit of info about this phenomenon.
Is “춥다
” an irregular adjective?
Yes, along with other “-ㅂ
” stem adjectives/verbs except for some. We can't say every such adjective/verb is, but a bunch are.
ㅂ irregular conjugation
It's called “ㅂ irregular conjugation (ㅂ 불규칙 활용).” A conjugation where a stem's final “ㅂ
,” as in “ㅂ
” in “춥-
,” becomes an “ㅗ
” or “ㅜ
” before a conjugation with no initial consonant, such as “-었-
.” E.g. “춥다
” → “추웠다
” (“추
” + (“ㅂ-
”→“ㅜ-
”) + “-었-
” + “-다
”).
Why does this happen?
Such a “ㅂ
” was originally a “ㅸ
” (unused nowaday), which likely had the /v/ sound in initial and the /ㅂ/ sound in final. Later, the /v/ sound shifted to the /w/ sound.
(“추ᇦ-
” + “-어
”) → “추ᄫᅥ
” (Since “-어
” has no initial consonant, the “ㅸ
” takes the initial position here.) → “추워
”
Fun fact: some dialects still have this sound.
The Kyungsang dialect. They say “추ᄫᅳ라 /choovɾa/” for the standard equivalent “추워라 /choowoɾa/.”