Exactly ! A full frame sensor has 33% less DOF than APS-C but receives 136% more light.Do you understand that the amount of light gathered on a FF sensor through f4.5 is equal to that on an APSC sensor through f2.8? So when stopping down to achieve the same DOF, the FF camera does not receive less light than the APSC camera.
APS-C crop factor is 1.5 so it has 50% more DOF than a FF sensor and receives 42% of its light.
When you stop down a lens by 1 stop you gain approximatively 1.45 X more DOF.
So at equivalent DOF, the FF sensor receives about 16% more light than an APS-C sensor.