Is there a word for "coolness" that corresponds to warmth?
There's the archaic word coolth:
(archaic) The state of being cool, temperature-wise; coolness. [eg] The water pushed large blocks of tepid air about around his chair, giving the faint illusion of freshness and coolth. – Lawrence Durrell, Constance, 1982
Edit 2: In many uses coolth corresponds better to warmth than does cool. Architects' use of coolth as mentioned in comments provides an example. Also, while cool corresponds well with warm, it does not correspond well with warmth or warmness in some examples below, shown in several forms for contrast:
His cool manner ... – ok
His warm manner ... – ok
His coolness of manner ... – ok
His warmth of manner ... – ok
*His cool of manner ... – bad
The coolth of his manner ... – ok
The coolness of his manner ... – ok
The warmness of his manner ... – ok
The warmth of his manner ... – ok
*The cool of his manner ... – bad
The chill of his manner ... – ok
Edit 1: Regarding whether coolth is archaic, Peter Shor notes that it appears in The Spectator (ca. 1776). It also appears in Fanny Burney memoirs, ca. 1784, and according to ngrams is rare before the late 1800's, and at its most-common from 1920 to 1960.
The most-relevant sense of archaic is "(of words) No longer in ordinary use, though still used occasionally to give a sense of antiquity." Coolth is still in use, but not in ordinary use; and it is, per previous paragraph, not particularly antique; but because it may often be used to give a sense of antiquity, it's reasonable to term it archaic.
The warmth of the sun versus the coldness of space.
I also note that the Merriam Webster's link for cold list definition 2a as
marked by a lack of the warmth of normal human emotion
so that would lend a lot to the answer of cold by @zpletan, but the sentence above is how I recall hearing it contrasted.
The short answer is no, there is not a single word commonly in use that is to "cool" what "warmth" is to "warm". "Coolth" just never caught on the way "warmth" did.
"Warm" and "warmth" both refer to a quality of temperature and a quality of friendliness and compassion. "Hot" is a higher temperature than "warm" but not really a higher degree of friendliness; "hot" can be either erotic or angry, but not "more friendly than warm". I believe as a result of this and because "warmth" has positive associations, "warmth" gets used a lot in both senses in preference to any other word with similar meaning, such as "warmness". Besides, "warmth" just sounds better. On top of that, with "warmth" readily available, "warm" did not develop a usage as a noun. (Then again, "hot" never developed into a noun, either, and instead we have "heat".)
On the other side, "cool" and "cold" both refer to a quality of temperature and a quality of friendliness and compassion and "cold" is "more cool" in both senses. Without "coolth" coming into use, "cool" and "cold" both became nouns distinct from "coolness" and "coldness".
In the above ngram graph from Google, "warm of day" does not even register when compared to "warmth of day", "cool of night", or "coolness of night".
Here is a comparison of the words alone. Most used is "cold", followed by "warm", "cool", and "warmth" in that order. Least used by far are "coolness" and "coldness".
On the subject of affection and matters of the heart, Google says "warmth" and "coldness" are most used by far when referring to someone's heart, with "warm", "cool", "coolness", and "cold" not registering at all:
(Yes, this graph shows usage of "her" heart, but the graph of "his" heart is pretty much the same.)
So you have to work a little harder to figure out the sense of what you want to say and choose the right word to mean the opposite of "warmth".