To my poor knowledge on the topic, the gravitational waves that are most likely to be detected by LIGO or other experiments do not have thermal spectrum. But I'm not certain.
I know that Hawking's radiation is thermal and it is partially make up of gravitational radiation. However, as far as I know (correct me), its detection is not possible in practice.
I think that the gravitational wave background is thermal, but I have no idea about its direct or indirect detection. Will (Advanced) LIGO be able to say anything about this? Maybe the B modes of CMB can tell us something?
Any information about possible sources of gravitational radiation with thermal spectrum will be welcome.