The most common and most likely answer to this crossword clue is the 3 plus 5 letter word ICE HOUSE
I
C
E
H
O
U
S
E
Buddy explains!

The straight part
We know that "cold in here" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case cold in here = ice house.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does ICE HOUSE mean?
ICE HOUSE noun- A deep cellar or outdoor building used for the storage of ice or snow; sometimes also used to store food at low temperature.
- An ice hockey rink.
- A cold state in global climate.