The nuclear continuum may potentially be investigated by means of fusion-evaporation reactions in combination with the $Hk$ technique. The $\ensuremath{\gamma}$ radiation in such experiments can be analyzed using energy ordering which produces complex spectra. A method for computing the shape of the energy-ordered spectra is presented and benchmarked by reproducing simulated data. The spectrum shape computation is applied to experimentally relevant cases and is used for extracting the level density parameter from simulations. Furthermore, the calculation shows to be useful for the characterization of nuclear phase transitions.