  The inflationary model proposed by Starobinski in 1979 predicts an amplitude
of the spectrum of primordial gravitational waves, parametrized by the tensor
to scalar ratio, of $r=0.0037$ in case of a scalar spectral index of
$n_S=0.965$. This amplitude is currently used as a target value in the design
of future CMB experiments with the ultimate goal of measuring it at more than
five standard deviations. Here we evaluate how stable are the predictions of
the Starobinski model on $r$ considering the experimental uncertainties on
$n_S$ and the assumption of $\Lambda$CDM. We also consider inflationary models
where the $R^2$ term in Starobinsky action is generalized to a $R^{2p}$ term
with index $p$ close to unity. We found that current data place a lower limit
of $r>0.0013$ at $95 \%$ C.L. for the classic Starobinski model, and predict
also a running of the scalar index different from zero at more than three
standard deviation in the range $dn/dlnk=-0.0006_{-0.0001}^{+0.0002}$. A level
of gravitational waves of $r\sim0.001$ is therefore possible in the Starobinski
scenario and it will not be clearly detectable by future CMB missions as
LiteBIRD and CMB-S4. When assuming a more general $R^{2p}$ inflation we found
no expected lower limit on $r$, and a running consistent with zero. We found
that current data are able to place a tight constraints on the index of
$R^{2p}$ models at $95\%$ C.L. i.e. $p= 0.99^{+0.02}_{-0.03}$.
