Both teams had already sealed a place in the quarter-finals but it was the French side who prevailed at Kingsholm, finishing the pool stage with a perfect record of six wins from six.
Billy Twelvetrees cut the gap to three points five minutes into the second half, however, any hope of a turnaround was extinguished as Pau ran in three more tries.
Jake Polledri went over late on for Gloucester but his score served as nothing more than a consolation.
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