  We generalize the Endo formula originally developed for the computation of
the heat kernel asymptotic expansion for non-minimal operators in commutative
gauge theories to the noncommutative case. In this way, the first three
non-zero heat trace coefficients of the non-minimal U(N) gauge field kinetic
operator on the Moyal plane taken in an arbitrary background are calculated. We
show that the non-planar part of the heat trace asymptotics is determined by
U(1) sector of the gauge model. The non-planar or mixed heat kernel
coefficients are shown to be gauge-fixing dependent in any dimension of
space-time. In the case of the degenerate deformation parameter the lowest
mixed coefficients in the heat expansion produce non-local gauge-fixing
dependent singularities of the one-loop effective action that destroy the
renormalizability of the U(N) model at one-loop level. The twisted-gauge
transformation approach is discussed.
