  In this study, we investigate the thermodynamics of a relativistic ideal
within the context of $\kappa$-deformed space-time and Rainbow gravity
background. To achieve this, we construct a modified partition function by
considering a deformed Hamiltonian and incorporating corrections based on the
time-invariant phase-space volume. We explore the implications of our model on
the modified black body radiation spectrum and the modified Debye theory of
specific heat in $\kappa$-deformed space-time and Rainbow gravity background.
