If it isn't a shootout the goalie can do whatever he wants
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No. Throwing your stick in any attempt to hit the ball is against the rules, and is penalised very severely in every form of hockey.
A goal keeper may throw the ball and score a goal. In practice such an occurrence is unlikely and would require a small field, a lot of wind, and an inattentive defense.
no you cant that's not allowed in hockey you have to hit it in with the hockey stick