I am not sure how this has passed me by but the Chinese Tea House in Frederiksberg Have is open every Sunday in the summer. It is a beautiful place to visit and enjoy a drink, made even more special as it is a rare thing. The 19th century pavilion and the island on which is sits is open May to September on Sunday afternoons.
Chinese Pavilion in Frederiksberg Have
It is a little crumbly and faded inside but I think that makes it even lovelier.