During your stay, I recommend taking the metro to explore. The metro/subway now makes it really fast and convenient (yellow metro line) to reach downtown Copenhagen city centre from the airport (Københavns Lufthavn). The metro goes every few minutes and will take 20 minutes to reach city centre from the airport. With a layover of more than 6-8 hours, there is definitely enough time to get a taste of Copenhagen. Great places to stop on the yellow metro line to experience Copenhagen are Christianshavn, Kongen Nytorv, Nørreport, Forum, and Frederiksberg. Just get off the metro, walk around and get a bite to eat instead of pre-packaged airport food.