A continuation of the Born Reciprocal Relativity Theory (BRRT) program in phase space shows that a natural temperature-dependence of mass occurs after recurring to the Fulling-Davies-Unruh effect. The temperature dependence of the mass m(T) resemblances the energy-scale dependence of mass and other physical parameters in the renormalization (group) program of QFT. It is found in a special case that the effective photon mass is no longer zero, which may have far reaching consequences in the resolution of the dark matter problem. The Fulling-Davies-Unruh effect in a \( D = 1 + 1\) -dim spacetime is analyzed entirely from the perspective of BRRT, and we explain how it may be interpreted in terms of a linear superposition of an infinite number of states resulting from the action of the group U(1, 1) on the Lorentz non-invariant vacuum \( \ | {\tilde{0}} \rangle \) of the relativistic oscillator studied by Bars [1].