  Unparticle $\U$ with scaling dimension $d_\U$ has peculiar thermal properties
due to its unique phase space structure. We find that the equation of state
parameter $\omega_\U$, the ratio of pressure to energy density, is given by
$1/(2d_\U +1)$ providing a new form of energy in our universe. In an expanding
universe, the unparticle energy density $\rho_\U(T)$ evolves dramatically
differently from that for photons. For $d_\U >1$, even if $\rho_\U(T_D)$ at a
high decoupling temperature $T_D$ is very small, it is possible to have a large
relic density $\rho_\U(T^0_\gamma)$ at present photon temperature $T^0_\gamma$,
large enough to play the role of dark matter. We calculate $T_D$ and
$\rho_\U(T^0_\gamma)$ using photon-unparticle interactions for illustration.
