Treat the pooch like a person. If it is relaxed in the plane, don't worry. If it is worried, better head down.
I was flying the boss's 172 once, with he, and his great dane aboard. She was a lovely dog, though perhaps simply lonely in the back. She tried to get into the front with us part way through the trip. The boss fought the battle, I flew the plane. I think I remember some yelling. I'm glad the upholstery repair bill did not come to me!