The specific heat corresponding to systems with deterministic fractal energy spectra is known to present logarithmic-periodic oscillations as a function of the temperature T in the low T region around a mean value given by a characteristic dimension of the energy spectrum. In general, it is considered that the presence of disorder does not affect strongly these results, and that the fractal structure of the energy spectrum dominates. In this paper, we study the properties of the specific heat derived from random fractal energy spectra as a function of the degree of disorder present in the spectra. To study the influence of the disorder, we analyze the specific heat using three different properties: the specific heat mean value and the periods and amplitudes of the oscillations of the specific heat around its mean value. By studying the distributions and the mean values of these three properties, we obtain that the disorder does not influence very much the mean value of the specific heat. However, concerning the behavior of periods and amplitudes, we obtain a critical value of the disorder present in the energy spectra. Below this critical value, we find a low effect of the disorder and quasideterministic behavior indicating that the fractal structure is the dominant effect, but above the critical value, the disorder dominates and the behavior of the specific heat is practically chaotic.