  Invariant tensors are states in the (local) SU(2) tensor product
representation but invariant under global SU(2) action. They are of importance
in the study of loop quantum gravity. A random tensor is an ensemble of tensor
states. An average over the ensemble is carried out when computing any physical
quantities. The random tensor exhibits a phenomenon of `concentration of
measure', saying that for any bipartition, the expected value of entanglement
entropy of its reduced density matrix is asymptotically the maximal possible as
the local dimension goes to infinity. This is also true even when the average
is over the invariant subspace instead of the whole space for $4-$valent
tensors, although its entropy deficit is divergent. One might expect that for
$n\geq 5$, $n-$valent random invariant tensor would behavior similarly.
However, we show that, the expected entropy deficit of reduced density matrix
of such $n-$valent random invariant tensor from maximum, is not divergent but a
finite number. Under some special situation, the number could be even smaller
than half a bit, which is the deficit of random pure state over the whole
Hilbert space from maximum.
