Star Alliance, Oneworld, and Icelandair have flights on sale from the East Coast of the US to Scandinavia starting at $309 round trip from Orlando and starting at $352 round trip from other East Coast cities in late August and September/October (and possibly later into the schedule). Skyteam has a couple of routes on sale as well. Also, Icelandair theoretically allows a free stopover in Iceland on the way. Some examples (click images to go to Google Flights):
New York to Stockholm, September 5-12:
The Icelandair flights can be booked with a free stopover in Iceland, though you may need to play with dates to do so. Here’s an example:
New York to Copenhagen, September 5-12:
Boston to Copenhagen, September 5-12:
Boston to Stockholm August 23rd-September 5th:
Philadelphia to Copenhagen is available for a little bit more (and also a bit earlier in August). Here is August 15-22:
Secret Flying is also reporting flights from Orlando to Copenhagen is on sale from $309 round trip. I can’t find that price via Google Flights, but it is available via Momondo. See the Secret Flying post for more example dates.
Orlando to Copenhagen September 10-21:
While the weather will start to cool off in September, these prices are still quite good for shoulder season — when hotels will likely be more reasonable and crowds thinner. Additionally, there will be Choice Hotel points coming up for sale in Daily Getaways. Remember that some Nordic Choice Hotels can offer great value for those points — including complimentary breakfast and dinner in some locations, making for a cheap getaway to Scandinavia. Bookings made through online travel agencies should have 24-hour free cancellation, but remember that if you are booking directly through the airline website that policy may vary.