It has been recently claimed that cosmologies with a time dependent speed o
f light might solve some of the problems of the standard cosmological scena
rio, as well as inflationary scenarios. In this Rapid Communication we show
that most of these models, when analyzed in a consistent way, lend to larg
e violations of charge conservation. Thus, they are severely constrained by
experiment, including those where c is a power of the scale factor and tho
se whose source term is the trace of the energy-momentum tensor. In additio
n, early universe scenarios with a sudden change of c related to baryogenes
is are discarded.