I have been there in August 2007.

Outside temperature was really high and even getting out from the car was a relief.
The tour inside the cave takes about 45 minutes, if I'm not mistaken, and it is organized in groups, so you could wait some minutes before getting in.
I personally really love cold and getting inside there was a freezing Valhalla for me, but mind-healthy people would better wear something warm.
Outside the entrance a sign was advising "t в переще от -3 до +5", which means "temperature inside the cave from -3 to +5" (Celsius degrees). Personally I was wearing a short sleeve shirt and a shorts, but I was the only one so undressed.
Funny anecdote: the bottle I had in the wide pocket of my shorts was chilly in the end of our tour.

The place is marvellous and in spite of the outside temperature there really was some ice inside a long path that water digged inside the rocks.
There are a lot of narrow passages, some of which are also quite slippery, connecting bigger grottos to each other. Evey grotto has a name that comes from a tale that it would be explained to you byt the guide. Closer to the end you can also see the water stream that created this natural jewel.
The guide explains everything about the place, but unfortunately, only in Russian. If you know the language or someone can translate for you it really worths it. I personally dream to be back there as see how it is in winter time. Temperature of course will be lower and you should wear the heavy clothes needed also to move outside. I would expect the area to change dress, as well, wearing an icy wintry one.