Any ideas for a temporary soundproofing solution?

Maybe you could buy a dense piece of foam and have it cut to exactly fit this gap. What I like about foam is that if it falls, it is a very soft material and will not hurt anyone or even the surface of your floor. Would you need to remove and replace the material each time the door was opened?

They use this type of material to restuff pillows and the like. Maybe a fabric store like Zarin's would have a good type of foam to use in a children's room.

http://www.zarinfabrics.com/Inserts-C195.aspx

https://www.google.com/search?q=site:streeteasy.com+soundproofing+neighbor

Lobster, no need to move the material to operate the door. Will look into Zarin's now. thanks for the input.

I would use a piece of rigid foam insulation - they come in 2 x 8 sheets usually (24" x 96"). You would need to paint it as they are generally bright pink. Would only cost you about $20. Pretty easy to cut to size with a utility knife.