No work no travel

If you're out of work in Denmark and you want to keep your unemployment benefits, you're basically stuck in it ... you're supposed to be looking for a job and apparently Denmark is the only place where you're allowed to look! No traveling abroad if you're jobless at home ... this seems to be the idea behind the random checks at airports (among other places) in Denmark by the Labour Directorate. In reality, this is not exactly true. You can still travel as long as you notify beforehand the Labour Directorate and get its approval. Traveling is allowed for family reasons, health issues and job searching. All reasons have to be proven by documentation.

In addition, while unemployed you still have right to holidays (!) ... having nothing to do can be very laborious after all. You'll have three weeks where you can travel without having to specify reasons. You'll still need to notify the Labour Directorate though!
If you get stopped at the airport by the Police, they will check on your labour situation and eventually let you know if they've opened an investigation about you.

This is the paper they will release to let you know what's going on:

Dansk
Arbejds Direktoratet
Arbejdsdirektoratet arbejder sammen med Politi og Skat i lufthavnen.
Vi kontrollerer, at du ikke modtager sociale ydelser uretmæssigt under ophold i udlandet.
Hvis direktoratet ikke finder grundlag for at undersøge dine forhold nærmere, vil du ikke høre mere fra os og dine personlige oplysninger bliver slettet.
Her kan du læse om persondataloven og dine rettigheder.

English
Labour Directorate
Labour Directorate works with the Police and the Tax Authority at the airport.
We make sure you do not receive social security benefits erroneously while staying abroad.
If the Directorate has no basis to investigate your situation further, you will not hear more from us and your personal information will be deleted.
You can read the Privacy Act and your rights (on the back of the paper).