It's really going to depend where you are trying to go. The town of Breck is dog friendly, but at the ski resort it's likely to vary. Each hotel has their own pet policy so I would recommend verifying those independently.
Grand Colorado on peak 8 has pet friendly rooms, I see dogs all the time on Richard's Run by the hot tubs. Theres also an area for them to use the bathroom on the top of the hill above the hot tubs.
You can call them up and check. I didn't see any Pets around while skiing. You should call them.
Thank you for the answers! Michelle thanks I realized that I meant to ask this about the Grand Colorado so Thank you!
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