In the fixed-field (cold) approximation, a generalized formula is derived for the starting currents of eigenmodes excited by an electron beam in a gyrotron cavity with the mode conversion. Tapered gyrotron cavities with gradual and abrupt transitions are considered as examples. It is shown that the effect of the mode conversion on the starting current consists mainly in the change of the diffractive quality factor and, therefore, is small for cavities with low diffractive losses. For such cavities, our formula is in good agreement with single-mode, self-consistent (hot) calculations of the electron beam-wave interaction. The agreement degrades for high-order axial modes having clear-cut distinctions of cold and hot fields. The derived formula is used to test the validity of the single-mode approximation for a complex gyrotron cavity with the abrupt change in radius.