Alaska is mostly too far north and outside of mountain lion distribution. As such, there isn't much of a resident population aside from a few cats that live in the southeastern edge of the state adjacent to populations living in Britis Columbia.
Climate change may be increasing the habitat available to mountain lions, only time will tell.
Alaska is north of mountain lions distribution for the most part, and cougars are rare in neighboring northern British Columbia. In southern B.C., biologists estimate the population to be about 3,500 individuals. Some of these cats may disperse to the north and to Alaska, but the likelihood of there being a breeding population within the state is very low.
This may seem like a good thing for mountain lions, but the story is not that simple. The complex network of relationships that exist in any ecosystem has spent a very long time evolving together. These associations develop very slowly over time, they are constantly changing and responding to one another. Given time, they could potentially adapt to a warmer climate and different weather patterns. However, the pace of climate change does not allow for slow, incremental development happen. Climate change affects some elements of the ecosystems faster than others and it may influence some aspects in very different ways than others, eventually potentially unraveling the important relationships that have coevolved over millennia.