Based on the scheme of "hot" trapped-ion quantum computation proposed by Sφrensen and Mφlmer, we construct a set of multi-mode excited logic gates on trapped ions, which may be operated on arbitrary even number of ions selected randomly from a string of ions trapped in a linear trap. The gate speed may be made faster than that of the corresponding one-mode excited logic gate by adjusting appropriately the detunings of driving lasers with respect to the sidebands of vibrational modes and can be demonstrated in current experimental techniques.