DTC P1065: Engine Control Module Power Supply (Back Up) Circuit : Description
WARNING: This page is about a different car, the 2003 Nissan Pathfinder. However, it is still accessible from the selected car via links, so may be relevant.
Battery voltage is supplied to the ECM even when the ignition switch is turned OFF for the ECM memory function of the DTC memory, the air/fuel ratio feedback compensation value memory, the idle air volume learning value memory, etc.