An electric power steering apparatus comprising a motor for providing a steering mechanism with steering assist force and a control unit for controlling the drive current of said motor, whereby said control unit comprises: a power self-holding means which can control own power off after power is once turned on by an IG switch; a temperature estimation means for estimating a temperature of said motor and performing a temperature control by the estimated temperature; a temperature estimation status storage means, in a case that a temperature estimation processing continues by said power self-holding means, for storing predetermined data indicating a status of said temperature estimation processing in a storage section within said control unit before said temperature estimation means stops; and an initial value setting means, when said IG switch is turned ON again and said temperature estimation means is activated, for setting an initial value of said estimated temperature based on said predetermined data indicating said status of said temperature estimation processing and a predetermined motor temperature estimation offset value, characteized by: said temperature estimation status storage means is a temperature stage storage means for storing a temperature stage to which said estimated temperature belongs in said storage section until said IG switch is turned OFF and said temperature stage reaches a predetermined stage when a steering assist by said motor ends and said temperature estimation processing continues by said power self-holding means, said initial value setting means reads out said motor temperature estimation offset value out of plural preset data and sets the initial value of said estimated temperature based on said temperature stage when said IG switch is turned ON again and said temperature estimation means is activated, and said temperature stage storage means stores said temperature stage in said storage section until said estimated temperature becomes a predetermined value or less, and also overwrites the present temperature stage in said storage section each time said temperature stage proceeds in a duration from at a time that said steering assist by said motor ends to at a time that said temperature estimation means stops. a power self-holding means which can control own power off after power is once turned on by an IG switch; a temperature estimation means for estimating a temperature of said motor and performing a temperature control by the estimated temperature; a temperature estimation status storage means, in a case that a temperature estimation processing continues by said power self-holding means, for storing predetermined data indicating a status of said temperature estimation processing in a storage section within said control unit before said temperature estimation means stops; and an initial value setting means, when said IG switch is turned ON again and said temperature estimation means is activated, for setting an initial value of said estimated temperature based on said predetermined data indicating said status of said temperature estimation processing and a predetermined motor temperature estimation offset value, characteized by: said temperature estimation status storage means is a temperature stage storage means for storing a temperature stage to which said estimated temperature belongs in said storage section until said IG switch is turned OFF and said temperature stage reaches a predetermined stage when a steering assist by said motor ends and said temperature estimation processing continues by said power self-holding means, said initial value setting means reads out said motor temperature estimation offset value out of plural preset data and sets the initial value of said estimated temperature based on said temperature stage when said IG switch is turned ON again and said temperature estimation means is activated, and said temperature stage storage means stores said temperature stage in said storage section until said estimated temperature becomes a predetermined value or less, and also overwrites the present temperature stage in said storage section each time said temperature stage proceeds in a duration from at a time that said steering assist by said motor ends to at a time that said temperature estimation means stops. said temperature estimation status storage means is a temperature stage storage means for storing a temperature stage to which said estimated temperature belongs in said storage section until said IG switch is turned OFF and said temperature stage reaches a predetermined stage when a steering assist by said motor ends and said temperature estimation processing continues by said power self-holding means, said initial value setting means reads out said motor temperature estimation offset value out of plural preset data and sets the initial value of said estimated temperature based on said temperature stage when said IG switch is turned ON again and said temperature estimation means is activated, and said temperature stage storage means stores said temperature stage in said storage section until said estimated temperature becomes a predetermined value or less, and also overwrites the present temperature stage in said storage section each time said temperature stage proceeds in a duration from at a time that said steering assist by said motor ends to at a time that said temperature estimation means stops.