The most common and most likely answer to this crossword clue is the 9 letter word MIDWINTER
I've seen this crossword clue in The Guardian
M
I
D
W
I
N
T
E
R
More clues leading to the result MIDWINTER
Buddy explains!

The straight part
We know that "the coolest of times?" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case the coolest of times? = midwinter.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 MIDWINTER mean?
MIDWINTER noun- The middle of winter.
- The winter solstice; about December 21st or 22nd.