I've always wanted to do a late December, early January Chamois hunt. The skins will be the summer 'colours' and supposedly they will be higher up the mountains. At least this is what this armchair chamois hunter has read.

At least in the mountains it will be cool as that time in the Southern Hemisphere will be mid-Summer.

Some red stags might be in hard antler by mid-Feb at least they are locally.

Have been doing just that of late.
Have just returned from Malbourough.
(Molesworth Station to be exact.)
Shot a nice fat spiker a couple of days ago, his velvet just starting to harden. (Shot him with my .44 Magnum) Saw 25 deer in total, two stags were 12 points, their tops grown out but still in velvet. Saw a nice Buck Chamois, but too far away for a shot, and a huge Ravine blocking a closer stalk. The summer skins are great, and make great headskins for trophies. Recently saw a combo mount, one winter and one summer looked great.
The biggest plus with this time of year is, very little snow and lots of daylight. Also not too many other hunters kicking about.
As for cool, I don't think so, damn hot, well for us kiwis (25 Celcius) and I have the sunburnt lips to prove it. The deer were very high 1200-1500 metres ASL and the Chamois surprisingly low 800 ASL
Also saw a heap of goats, but not a single pig nor fresh pig sign (Unusual for the area.)