You should do both, calculate and note each type of damage, but also apply the damage as a whole.
Most of the time, it won’t make a difference, because most creatures lack damage resistances or weaknesses. But of the creatures that do resist damage, they may have resistance to one type of damage, but not the other. Take the adult red dragon as an example. It has weakness to cold 15, and no special weakness or resistance to slashing. If you hit it with your axe for 8 slashing and 4 cold, it takes the 8 slashing, and increases the cold damage by its weakness value (a total of 19 cold damage).
At the same time, if you have to make a check based on the total amount of damage dealt, then you don’t make 2 checks, one for the slashing damage and one for the cold damage, you make one check based on the total of 27.
So the damage needs to be separated when applying it to weaknesses and resistances, then apply the final result as a single instance of damage.