is my dog's white paw more sensitive to cold?

If this is a new dog and the dog has previously not gotten a lot of exercise it may be too much exercise for him right now. He may need to ease into it and also he could've had a previous injury to that particular leg or foot that is now causing him some stiffness along with the overexertion. And because he may have a previous injury the cold will aggravate it along with the overexertion. 

 Maybe take shorter hikes or walks until he gets into better shape. You can ease the stiffness with some massage.

 For my own dog I massage him along with using essential oils by Young Living - mixed with their massage oil Ortho-Ease. I also may use a a dose of Arnica Montana 30C (homeopathic remedy) to ease the soreness - and of course that's just what I do for my own dog. 

Thank you, Dr. Kim.  That's a wonderful and thorough response.  Way beyond what I expected when I sent my question in.  To clarify, this was certainly a walk further than he has ever gone before.  I am just sure of that.  And your thinking is that the cold may have aggravated an old injury.  The fact that it is his white paw is coincidence and any injury or discomfort would likely be indirectly rather than directly related to the cold?  Forgetting the fact that he could have actually hurt himself, of course;-)

Just to add to what Kim said, I wouldn't expect the white color in and of itself to cause a greater sensitivity. Just think about all those white animals who live in cold climates, like polar bears or Samoyeds who were bred to work there, plus those species like arctic foxes who turn white every winter for better protection. :-)

Also a few other thoughts relative to injury, sometimes dogs who have problems with both hips or knees may put more pressure on their front legs than usual. In that case, even a small injury to a front leg may cause an exaggerated response. Because dogs who have been spayed or neutered very young are more prone to such rear end problems, this is something else to consider if the front end problem recurs  or doesn't respond to treatment.Certain breeds are also more prone to elbow problems and that's another possibility.