Cougars are very active all through the year but from what I've seen myself cougars are the most active during cold winter months.
The time of the day when cougars are most active though is at night actually and not during the day.
Cougars are most active and out during the night hours of between dusk until dawn and this time when cougars are most active requires much extra vigilance by hikers while in any cougar country.
So if you're in cougar country where cougars are mostly you need to be very careful anytime during year and most careful during the dusk to dawn hours.
But also still be very careful if you're out in the woods and hiking, camping etc even during the day because cougars are also very active during the day as well.