  Quantum field theories (QFT's) on noncommutative spacetimes are currently
under intensive study. Usually such theories have world sheet noncommutativity.
In the present work, instead, we study QFT's with commutative world sheet and
noncommutative target space. Such noncommutativity can be interpreted in terms
of twisted statistics and is related to earlier work of Oeckl [1], and others
[2,3,4,5,6,7,8]. The twisted spectra of their free Hamiltonians has been found
earlier by Carmona et al [9,10]. We review their derivation and then compute
the partition function of one such typical theory. It leads to a deformed black
body spectrum, which is analysed in detail. The difference between the usual
and the deformed black body spectrum appears in the region of high frequencies.
Therefore we expect that the deformed black body radiation may potentially be
used to compute a GZK cut-off which will depend on the noncommutative parameter
$\theta$.
