  Primordial Gravitational Waves, i.e. a background of metric perturbations
sourced by the quantum inflationary fluctuations, if measured, could both
provide a substantial evidence for primordial inflation and shed light on
physics at extremely high energy scales. In this work we focus on their
propagating speed. Using an effective field theory approach we introduce a
time-dependent propagating speed $c_{\rm T}(t)$ showing that also small
deviations from the General Relativity (GR) prediction $c_{\rm T}(t) = c$ can
lead to testable consequences. We derive a set of equations that relate the
propagating speed and its time dependence to the inflationary parameters and
that generalize the usual slow roll consistency relations. Imposing the new
generalized consistency relations and combining small and large scales data, we
derive model independent constraints on inflation with non-trivial primordial
tensor speed. In particular we constrain its scale dependence to be $d\log
c_{\rm T} / d\log k=0.082^{+0.047}_{-0.11}$ at 68% C.L. while we only derive
the lower bound $c_{\rm T}>0.22\,c$ at 95% C.L. . We also constrain the
tensor-to-scalar ratio at the pivot scale $k_*=0.05\rm{Mpc}^{-1}$ to be
$r<0.0599$ at 95% C.L. in agreement with the result provided by the Planck
collaboration. Thanks to a proper small scale parameterization of the tensor
spectrum we derive stringent constraints on the tensor tilt $n_{\rm
T}=-0.084^{+0.10}_{-0.047}$ at 68% C.L. and on its runnings $\alpha_{\rm
T}=d\,n_{\rm T}/d\log k=0.0141^{+0.0035}_{-0.021}$ and $\beta_{\rm
T}=d\,\alpha_{\rm T}/d\log k= -0.0061^{+0.010}_{-0.0014}$ both at 68% C.L. Our
results show a remarkable agreement with the standard slow roll predictions and
prove that current data can significantly constrain deviations from GR on the
inflationary energy scales.
