Both 'featherbed' and 'feather bed' are correct. While 'feather bed' is more common, anytime that the term is used in a figurative sense (i.e., not referring to a mattress), 'featherbed' becomes more common.
1.
noun. Literally, a feather bed is a bed in which the mattress is stuffed with feathers. They have a reputation for being very soft.
2.
verb. To hire more workers than are really necessary. This might be done because a union requires it, or to satisfy safety requirements. See featherbedding, featherbed rules.
3. verb (rare)
To pamper or to treat with excessive indulgence.
4. noun. A fibrous layer of living plants and organic material floating on top of a pond (or other body) of water.