Thermal properties of Klein–Gordon oscillator in the context of Amelino-Camelia and Magueijo–Smolin doubly special relativity (DSR) frameworks

Abstract We examine the thermal and statistical properties of the one-dimensional Klein–Gordon oscillator within two prominent Doubly Special Relativity (DSR) frameworks: Amelino-Camelia and Magueijo–Smolin. Using the modified dispersion relations specific to each formulation, we derive the positive energy spectra, construct the partition function via the Euler–Maclaurin method, and compute key thermodynamic quantities, including the specific heat Cv, as functions of temperature and the deformation scale. Planck-scale corrections produce distinct, theoretically resolvable shifts in both the position and magnitude of the Cv peak in the two models. An accompanying entropy analysis reveals that these peaks correspond to smooth Schottky-type anomalies: the specific heat curves remain analytic and positive across the explored temperature range, and thus do not indicate latent or continuous thermodynamic phase transitions. These comparative results provide a robust diagnostic framework for differentiating DSR prescriptions in relativistic quantum systems and reinforce the transition-free character of their thermal response.