Li2B4O7 activated with a Cu impurity exhibits the 'unique' characteristics of linear thermoluminescence (TL) effects, a rare feature very similar to that of LiF:Mg, Cu, P, another phosphor of this category. Its thermoluminescence is investigated for its trap distribution characteristics. The analysis, based on a fractional glow technique, shows that TL processes in Li2B4O7:Cu involve a number of trap groups with activation energies in the range of 0.9 to 1.73 eV distributed 'conspicuously' in the 80-355 degrees C glow temperature region, strikingly dissimilar to that of other common TL phosphors. The main glow peak region of 170-260 degrees C with a peak at 220 degrees C has been linked to the presence of predominantly deep trapping levels having energy in the region of 1.65+or-0.08 eV. The adjoining lower- and higher-temperature TL contributions are ascribed to the occurrence of shallower levels with considerably lower activation energies of thermal stimulation. These thermoluminescence characterisation data, in terms of the competing TL trap distribution model proposed earlier, satisfactorily explain the 'unique' behaviour of this material. The common and exceptionally rare TL features of Li2B4O7:Cu and LiF:Mg,Cu,P are attributed to the physical characteristics of Cu as a dopant. It is suggested that doping with Cu introduces predominantly deeper trapping levels in lattices like Li2B4O7 and LiF.