We introduce the theory of thermodynamic relativity, a unified framework for describing both entropies and velocities, and their respective disciplines of thermodynamics and kinematics, which share a surprisingly identical description with relativity. This is the first study to generalize relativity in a thermodynamic context, leading naturally to anisotropic and nonlinear adaptations of relativity; thermodynamic relativity constitutes a new path of generalization, as compared to the traditional passage from special to general theory based on curved spacetime. We show that entropy and velocity are characterized by three identical postulates, providing the basis of a broader framework of relativity: (1) no privileged reference frame with zero value; (2) existence of an invariant and fixed value for all reference frames; and (3) existence of stationarity. The postulates lead to a unique way of addition for entropies and for velocities, called kappa addition. The theory provides a systematic method of constructing a generalized framework of the theory of relativity, based on the kappa addition, fully consistent with both thermodynamics and kinematics. From the generality of the kappa addition, we focus on the cases corresponding to linear special relativity. Then, we show how the thermodynamic relativity leads to the addition of entropies in nonextensive thermodynamics and the addition of velocities in the isotropic special relativity of Einstein, as in two extreme cases, while intermediate cases correspond to an anisotropic adaptation of relativity. Using thermodynamic relativity for velocities, we construct the anisotropic special relativity (asymmetric Lorentz transformation, nondiagonal metric, energy momentum velocity relations). Then, we discuss the consequences of the anisotropy in known relativistic effects: (1) matter antimatter asymmetry, (2) time dilation, and (3) Doppler effect.