  The situation with respect to the experiments is presented of a recently
proposed model that gives an explanation of the Pioneer anomalous acceleration
$a_{\rm P}$. The model is based on an idea already discovered by Einstein in
1907: the light speed depends on the gravitational potential $\Phi$, so that it
is larger the higher if $\Phi$. The potential due to all the mass and energy in
the universe increases in time because of its expansion, which has the
consequence that light must be slowly accelerating. Moreover it turns out that
the observational effects of a universal adiabatic acceleration of light
$a_\ell =a_{\rm P}$ and of an extra acceleration towards the
  Sun $a_{\rm P}$ of a spaceship would be the same: a blue shift increasing
linearly in time, precisely what was observed. The phenomenon would be due to a
cosmological acceleration of the proper time of bodies with respect to the
coordinate time. It is shown that it agrees with the experimental tests of
special relativity and the weak equivalence principle if the cosmological
variation of the fine structure constant is zero or very small, as it seems
now.
