  Our goal is to interpret the energy equation from Doubly Special Relativity
(DSR) of Magueijo-Smolin with an invariant Planck energy scale in order to
obtain the speed of light with an explicit dependence on the background
temperature of the expanding universe. We also investigate how other universal
constants, including the fine structure constant, have varied since the early
universe and, thus, how they have evoluted over the cosmological time related
to the temperature of the expanding universe. For instance, we show that both
the Planck constant and the electron charge were also too large in the early
universe. However, we finally conclude that the fine structure constant has
remained invariant with the age and temperature of the universe, which is in
agreement with laboratory tests and some observational data.
