Lava River Cave near Flagstaff- can I take my dog?

I can't find anywhere that says I can't take my dog... but I'd hate to get him out there and not be able to take him inside. I'm looking for some nice, chilly places to take a dog (he's a 4 month old, 20lb Golden Retriever) who is used to subzero temps at 11,000 feet, not Arizona temps! Any shady hikes with swimming? Hikes without a lot of slickrock? He's used to riding in my backpack, so if it's a long hike, no worries. Thanks in advance for being my tour guide!