Primordial gravitational waves, i.e., a background of metric perturbations sourced by the quantum inflationary fluctuations, if measured, could both provide 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 cT(t) showing that also small deviations from the general relativity (GR) prediction cT(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 nontrivial primordial tensor speed. In particular, we constrain its scale dependence to be dlogcT/dlogk=0.082-0.11+0.047 at 68% C.L. while we only derive the lower bound cT>0.22c at 95% C.L. We also constrain the tensor-to-scalar ratio at the pivot scale k∗=0.05 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 parametrization of the tensor spectrum we derive stringent constraints on the tensor tilt nT=-0.084-0.047+0.10 at 68% C.L. and on its runnings αT=dnT/dlogk=0.0141-0.021+0.0035 and βT=dαT/dlogk=-0.0061-0.0014+0.010 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
