Three of the cases were filed in Shariatpur and the two others in Netrakona and Jamalpur on Sunday.
Anam is accused in the cases of trying to uproot the democratic government system by helping the unconstitutional government in its attempt to banish Hasina and other political leaders from politics.
The cases are being filed against him after he admitted in a TV talk-show on Feb 6 that he ran the reports with unsubstantiated input from the military intelligence agency DGFI.
The first sedition case over these reports was filed in Dhaka on Thursday.
The Daily Star editor has also been sued in several defamation suits.
Hasina’s son Sajeeb Wazed Joy demanded that Anam to be tried for treason, a demand that was echoed by several MPs during a Parliament session.
The lawmakers also demanded that the newspaper be shut down.