  We study the effect of population inversion associated with the electron and
hole injection in graphene p-i-n structures at the room and slightly lower
temperatures. It is assumed that the recombination and energy relaxation of
electrons and holes is associated primarily with the interband and intraband
processes assisted by optical phonons. The dependences of the electron-hole and
optical phonon effective temperatures on the applied voltage, the
current-voltage characteristics, and the frequency-dependent dynamic
conductivity are calculated. In particular, we demonstrate that at low and
moderate voltages the injection can lead to a pronounced cooling of the
electron-hole plasma in the device i-section to the temperatures below the
lattice temperature. However at higher voltages, the voltage dependences can be
ambiguous exhibiting the S-shape.
  It is shown that the frequency-dependent dynamic conductivity can be negative
in the terahertz range of freqiencies at certain values of the applied voltage.
The electron-hole plasma cooling substantially reinforces the effect of
negative dynamic conductivity and promotes the realization of terahertz lasing.
On the other hand, the heating of optical phonon system can also be crucial
affecting the realization of negative dynamic conductivity and terahertz lasing
at the room temperatures.
