The government also argues he violated NDAs by giving speeches on intelligence issues. As such, it's seeking to recover all of Snowden's earnings from the book. The US won't try to hamper publication or distribution of Permanent Record, though -- doing so would violate Snowden's First Amendment rights.
It might seem difficult, on the surface, for the US to actually claim his proceeds, given Snowden is living in Russia under asylum. However, the government is suing the book's publisher as well, in an attempt to prevent earnings from being transferred to him. Should he ever return to the US, Snowden also faces criminal charges for allegedly disclosing classified information.