  In this work we shall provide a model-independent general calculation of the
running of the spectral index for vacuum $F(R)$ gravities. We shall exploit the
functional form of the spectral index and of the tensor-to-scalar ratio in
order to present a general $n_s-r$ relation for vacuum $F(R)$ gravity theories.
As we show, viable $F(R)$ gravity theories can be classified to two classes of
models regarding their prediction for the running spectral index. The
$R^2$-attractor models predict a running of the spectral index in the range
$-10^{-3}<a_s<-10^{-4}$, which classifies them in the same universality class
that most inflationary scalar field models belong to. We provide three models
of this sort, for which we verify our claims in detail. However there exist
viable $F(R)$ gravity models with running of spectral index outside the range
$-10^{-3}<a_s<-10^{-4}$ and in some cases it can be positive. We also present
an $R^2$-corrected scalar field model, which also predicts a running of the
spectral index in the range $-10^{-3}<a_s<-10^{-4}$. For all the cases we
studied, we found no evidence for the most phenomenologically interesting
scenario of having $r<10^{-4}$ and a running $a_s<-10^{-3}$, which in principle
could be realized.
