I don't think installing air channels would do much good if the soffits weren't vented.
Does the cathedral ceiling go all the way to the ridge? Or is there a little bit of attic space up there? (If so, can you install a fan anywhere?)
Could you consider a lighter shingle to reflect more heat?
Can the vaulted ceiling rafters be vented into the attic space of the "Normal" North/South roof?
..Those are the only ideas I can think of without seeing it ....
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