An optical medical instrument, in particular an endoscope or exoscope, has an elongate tubular shaft, a heat source generating lost heat, and a heat pipe which extends inside the shaft, in the longitudinal direction of the shaft, and has a distal heat pipe end and a proximal heat pipe end, wherein the heat pipe is thermally coupled to the heat source in order to collect heat from the latter and remove the heat from the heat source. Between the distal heat pipe end and the proximal heat pipe end, the heat pipe is coupled thermally conductively and two-dimensionally to the shaft over at least a partial length of the heat pipe and over at least a partial circumference of the heat pipe, in order to remove heat from the heat pipe, over at least a partial length and over at least a partial circumference of the shaft, to the environment.