This expression wasn't influenced by drowsy, hot puppies. According to National Geographic, the "dog" in question is Sirius, the Dog Star, which emerges in the sky in late July as a member of the Greater Dog constellation. The term "dog days" was used by the Greeks and Romans to describe the hottest part of the year, which was also thought to bring catastrophes like fever.
This was back when the couple was attempting to get the film made in Hollywood with only one puppet.
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