  The perfect fluid in the context of a covariant variable speed of light
theory proposed by J. Magueijo is studied. On the one hand the modified first
law of thermodynamics together with a recipe to obtain equations of state are
obtained. On the other hand the Newtonian limit is performed to obtain the
nonrelativistic hydrostatic equilibrium equation for the theory. The results
obtained are used to determine the time variation of the radius of Mercury
induced by the variability of the speed of light ($c$), and the scalar
contribution to the luminosity of white dwarfs. Using a bound for the change of
that radius and combining it with an upper limit for the variation of the fine
structure constant, a bound on the time variation of $c$ is set. An independent
bound is obtained from luminosity estimates for Stein 2015B.
