The dog in "To Build a Fire" is a husky-wolf mix, perfectly suited to life in the extreme cold. Its layered fur and natural instinct to burrow and conserve warmth are not shared by the man, who believes himself able to walk in the cold without proper preparations.
It experienced a vague but menacing apprehension that subdued it and made it slink along at the man's heels... as if expecting him to go into camp or to seek shelter somewhere and build a fire. The dog had learned fire, and it wanted fire, or else to burrow under the snow and cuddle its warmth away from the air.
(London, "To Build a Fire," eNotes eText)
The dog doesn't have the reasoning ability to understand temperature, but it does have the instinct to know when the cold becomes deadly. Since it is domesticated, it wants its human to provide it with fire, but even without fire it can survive. However, some instinctual part of the dog recognizes that the man is weaker against the elements, and it feels the connection strongly enough to want to herd the man to shelter. However, without the ability to reason, it can do little except walk alongside, waiting for the man to come to his senses.
The story is "To Build a Fire."