In order to diagnose a coolant pump (11) which is provided for the purpose of circulating a coolant in a closed cooling circuit of an internal combustion engine (10) and which can be activated and deactivated independently of the operating state of the internal combustion engine (10), both a value representing the coolant temperature (TCO) of the internal combustion engine (10) and a value representing the cylinder head temperature (TZK) of the internal combustion engine (10) are determined at a predefined time instant (t2) after a cold start of the internal combustion engine (10) has been detected and the values are subsequently compared with each other. The coolant pump (11) is rated in terms of its operational integrity as a function of the result of the comparison. This enables a faulty coolant pump to be detected at a very early stage after a cold start of the internal combustion engine.