We analyze the operation of the hot-electron FET bolometers with the graphene channels (GCs) and the gate barrier layers. Such bolometers use the thermionic emission of the hot electrons heated by incident-modulated THz radiation. The hot electrons transfer from the GC into the metal gate. As the THz detectors, these bolometers can operate at room temperature. We show that the response and ultimate modulation frequency of the GC-FET bolometers are determined by the efficiency of the hot-electron energy transfer to the lattice and the GC side contacts due to the 2DEG lateral thermal conductance. The dependences of these mechanisms on the band structure and geometrical parameters open the way for the GC-FET bolometers optimization, in particular, for the enhancement of the maximum modulation frequency.