The issur of tzar b’allei chaim prohibits causing unnecessary pain or suffering to an animal. The key word is “unnecessary”. For instance, shechting a cow certainly causes pain to the animal but the shechita is necessary and, therefore, muttar. Likewise, hunting for any halachically legitimate purpose should be permitted. While one can clearly not hunt for food (unless you could somehow capture a kosher animal alive, and shect it) if, say, you wanted the dear skin for a klaph, or a bear skin for a rug, or you wanted to shoot coyotes for their fur, It seems to me that such hunting would technically be muttar.
P.S. I don’t think some posters are clear about what “hunting for sport” is. To my knowledge, all big game hunting is for food in that the meat is never wasted. In America, almost all hunters are meat hunters, that is they butcher and eat their prey. On guided big game hunts in Alaska, the hunter can either keep the meat for him or herself or it gets distributed among the local villages. In Africa the meat is always distributed among the local population. All the hunter gets is the pelt.