A back-up system for an engine coolant temperature sensor detects deviation of the output of the temperature sensor from its designed output range to produce a fault signal. In response to the fault signal, an engine control system derives the engine coolant temperature indirectly from other engine conditions. This derivation is based on two facts: (1) the amount of fuel required to start the engine is related to engine temperature, and (2) since the calorific value of a given engine is essentially constant, the rate of increase of engine temperature is related to the integrated number of engine revolutions. Thus, when the engine is to be started, engine temperature can be derived by gradually increasing the fuel supply quantity from a minimal initial value until the engine is able to start to determine the required fuel supply quantity. Thereafter, the derived temperature value can be updated as a function of total engine revolutions.