  A way to address the conundrum of Quantum Gravity is to illustrate the
potentially fundamental interplay between quantum field theory, curved
space-times physics and thermodynamics. So far, when studying moving quantum
systems in the vacuum, the only known perfectly thermal temperatures are those
obtained for constant (or null) accelerations $A$ in constant (or null) Hubble
parameters $H$ space-times. In this Letter, restricting ourselves to
conformally coupled scalar fields, we present the most comprehensive expression
for the temperature undergone by a moving observer in the vacuum, valid for any
time-dependent linear accelerations and Hubble parameters: $T=\sqrt{A^2 + H^2 +
2 \dot H\dot t}/{2\pi}$ where $\dot t=\d t/\d\t$ is the motion's Lorentz
factor. The inequivalence between a constant $T$ and actual thermality is
explained. As a byproduct, all the Friedman universes for which observers at
rest feel the vacuum as a thermal bath are listed.
