Some implications of a scale invariant model of statistical mechanics to the mechanical theory of heat of Helmholtz and Clausius are described. Modified invariant definitions of heat and entropy are presented closing the gap between radiation and gas theory. Modified relativistic transformations of pressure, Boltzmann constant, entropy, and density are introduced leading to transformation of ideal gas law. Following Helmholtz the total thermal energy of thermodynamic system is decomposed into free heat U and latent heat p V and identified as modified form of the first law of thermodynamics Q = H = U + p V. Subjective versus objective aspects of Boltzmann thermodynamic entropy versus Shannon information entropy are discussed. Also, modified thermodynamic properties of ideal gas are presented. The relativistic thermodynamics being described is in accordance with Poincare - Lorentz dynamic theory of relativity as opposed to Einstein kinematic theory of relativity since the former theory that is based on compressible ether of Planck is causal as was emphasized by Pauli.