Using large-scale computer simulations, we thoroughly study the minimum energy required to thermally induced magnetization switching (TIMS) after the application of a femtosecond heat pulse in transition metal-rare earth ferrimagnetic alloys. We find that for an energy efficient TIMS, a low ferrimagnetic net magnetization with a strong temperature dependence is the relevant factor for the magnetic system. For the lattice and electron systems, the key physics for efficient TIMS is a large electron-phonon relaxation time. Importantly, we show that as the cooling time of the heated electrons is increased, the minimum power required to produce TIMS can be reduced by an order of magnitude. Our results show the way to low power TIMS by appropriate engineering of magnetic heterostructures.