We calculate cosmological and astrophysical bounds on the couplings between standard model fields and tensor unparticles. The present day density of tensor unparticles from neutrino-neutrino and photon-photon annihilation is calculated. Also, the supernovae volume energy loss rates from electron-positron and photon-photon annihilation to tensor unparticles are calculated. The constraints from matter density and supernovae volume energy loss rates from photon-photon annihilation are on the same order of magnitude, while the bounds from supernovae volume energy loss rates from electron-positron annihilation are an order of magnitude lower. We find the couplings between standard model fields and tensor unparticles are at least an order of magnitude lower than those used for previous studies of tensor unparticle collider phenomenology.