I have discussed this issue before - at the core, you have to ask yourself what is the carrier of the LENR energy? If many LENR events occur in a small NAE, and the energy carrier is phonon, then the temperature of the NAE will be the hottest with temperature falling off quickly with radius due to the small wavelength of the phonons. This would make the NAE burn out before significant energy is transferred to a lattice. On the other hand, if the energy is being carried away by low energy photons, they will not be immediately absorbed at the NAE, but can be absorbed in the condensed matter volume around the NAE. This could make the environment hotter than the NAE, allowing it to continue generating LENR events. It is like your microwave oven heating a potato - the potato can get hot without the plate the potato is sitting on. In this case the NAE is the microwave oven and the potato is the surrounding condensed matter. The heated volume in this case will depend on the energy of the photons being generated.
Of course, this argument applies to LENR being generated at NAE's as Ed Storms prescribes in his theory. Peter Haglestein suggests that phonons are involved in carrying away the heat. How does he address the temperature issue with the phonons? He says there are no NAE, but instead LENR happens one event at a time distributed in the condensed matter lattice. For example, a pair of D's fuse at a vacancy in the lattice but the next LENR event will occur somewhere else in the lattice. Peter's distributed LENR theory doesn't really seem to address the volcanoes that were seen peppering the surface of an active electrode - those don't seem to be distributed events, but rather more like a super-active NAE.