  We investigate the primordial gravitational waves (PGWs) in the general scenario where the inflation is preceded by a pre-inflationary stage with the effective equation of state $w$. Comparing with the results in the usual inflationary models, the power spectrum of PGWs is modified in two aspects: One is the mixture of the perturbation modes caused by he presence of the pre-inflationary period, and the other is the thermal initial state formed at the Planck era of the early Universe. By investigating the observational imprints of these modifications on the B-mode polarization of cosmic microwave background (CMB) radiation, we obtain the constraints on the conformal temperature of the thermal gravitational-wave background $T<5.01\times 10^{-4}$Mpc$^{-1}$ and a tensor-to-scalar ratio $r<0.084$ ($95\%$ confident level), which follows the bounds on total number of e-folds $N>63.5$ for the model with $w=1/3$, and $N>65.7$ for that with $w=1$. By taking into account various noises and the foreground radiations, we forecast the detection possibility of the thermal gravitational-wave background by the future CMBPol mission, and find that if $r>0.01$, the detection is possible as long as $T>1.5\times 10^{-4}$Mpc$^{-1}$. However, the effect of different $w$ is quite small, and it seems impossible to determine its value from the potential observations of CMBPol mission. 
Thermal gravitational-wave background in the general pre-inflationary   scenario