  A deconstructed finite temperature gauge theory has the Euclidean time
direction kept discrete and finite. The ultraviolet behavior is that of one
dimension less than at zero temperature. One can add to the action a gauge
invariant term dependent on Polyakov loops, without worrying about nonlocality.
The deformation of {\" U}nsal and Yaffe is best analyzed in this framework. It
is shown that turning it on causes a dramatic change in the Hilbert space on
which the transfer matrix in the temperature direction acts. The undeformed
Hilbert space is everywhere a gauge singlet while the deformed one includes all
states transforming locally, anywhere, under any SU(N)/Z(N) irreducible
representation. Implications of this in the context of large N reduction are
discussed.
