Federal Reserve Accountability Act of 2023 This bill changes the appointment of certain officials to the of Board of Governors of the Federal Reserve System and makes other changes to the operation of the board and Federal Reserve banks. Specifically, the bill requires the President to appoint a general counsel for the board with the advice and consent of the Senate; requires the President, rather than each bank's board of directors, to appoint presidents of Federal Reserve banks; reduces the number of Federal Reserve districts from 12 to 5; allows the President to appoint two members to the board from any one Federal Reserve district; and prohibits the use of any board or Federal Reserve bank funds for lobbying activities.