  The so-called unimodular version of General Relativity is revisited.
Unimodular gravity is constructed by fixing the determinant of the metric, what
leads to the trace-free part of the equations instead of the usual Einstein
field equations. Then, a cosmological constant naturally arises as an
integration constant. While unimodular gravity turns out equivalent to General
Relativity (GR) at the classical level, it provides important differences at
the quantum level. Here we extend the unimodular constraint to some extensions
of General Relativity that have drawn a lot of attention over the last years,
as $f(R)$ gravity (or its scalar-tensor picture) and Gauss-Bonnet gravity. The
corresponding unimodular version of such theories is constructed as well as the
conformal transformation that relates the Einstein and Jordan frames for these
non-minimally coupled theories. From the classical point of view, the
unimodular versions of such extensions are completely equivalent to their
originals, but an effective cosmological constant arises naturally, what may
provide a richer description of the universe evolution. Here we analyze the
case of Starobisnky inflation and compared with the original one.
