Apparatus for regulating the flow rate of fuel to a turboshaft engine in acceleration or in deceleration, the engine having a free turbine and a core engine, the apparatus comprising sensors transmitting information to regulator means, said information relating: to a first speed of rotation NTL of said free turbine; to a second speed of rotation Ng of said engine's gas generator; to an internal temperature T4 of the gas at the inlet to the free turbine; to the external pressure; and to the external temperature. The apparatus further comprises control means activated by said regulator means to actuate a fuel metering system of the engine. In addition, the regulator means evaluates the flow rate of fuel to be supplied to the engine in acceleration or in deceleration on the basis of at least one optimum regulation relationship, said regulation relationship determining a main modulated flow rate as a function of a modulated speed of rotation.