We investigate the thermodynamical properties of plasmon polaritons that propagate in multiple semiconductor layers arranged in a quasiperiodical fashion. This quasiperiodicity can be of the so-called deterministic (or controlled) disorder type, i.e., they are neither random nor periodic. Also, they are characterized by the nature of their Fourier spectrum, which can be dense pure point (Fibonacci sequence) or singular continuous (Thue-Morse sequence). The sequences are described in terms of a series of generations that obey peculiar recursion relations. We present both analytical and numerical studies on the temperature dependence of the polariton's specific heat associated with the generation number $n=1,2,3,\dots{}$ for their multiscale fractal energy spectra. We show that when $\stackrel{\ensuremath{\rightarrow}}{T}0,$ the specific heat displays oscillations and when $\stackrel{\ensuremath{\rightarrow}}{T}\ensuremath{\infty},$ the specific heat goes to zero with ${T}^{\ensuremath{-}2}$ (because the energy spectrum considered is bounded).