Should the energy-momentum tensor be invariant under gauge transformations? For example, consider the electromagnetic theory given by
\begin{align}
I=-\frac{1}{4}\int d^4x\, F_{\mu\nu}F^{\mu\nu},
\end{align}
where $F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu$. The action has a symmetry given by the space-time translations
\begin{align}
\delta A_\mu=-\epsilon^\alpha\partial_\alpha A_\mu,
\end{align}
with $\epsilon^\alpha$ the transformation parameter. In this case the energy-momentum tensor will be given by
\begin{align}
T^{\alpha}_\nu=F^{\mu\alpha}\partial_\nu A_\mu-\frac{1}{4}\delta^{\alpha}_\nu F_{\lambda\rho}F^{\lambda\rho}.
\end{align}
Everything seems to be ok, but note $T^{\alpha}_\nu$ will not be gauge-invariant and this is because $\delta A_\mu=-\epsilon^\alpha\partial_\alpha A_\mu$ is not gauge-invariant. This seems to be a problem, so one can do some sleight-of-hand and do an 'improved translation', which is a space-time translation + a particular gauge transformation
\begin{align}
\delta A_\mu=-\epsilon^{\alpha}\partial_\alpha A_\mu+\partial_\mu(\epsilon^{\alpha}A_\alpha)=F_{\mu\alpha}\epsilon^\alpha.
\end{align}
Note the transformation will be gauge-invariant. For completeness, the (gauge-invariant) energy-momentum tensor will be
\begin{align}
T^{\alpha}_\nu=-F^{\alpha\beta}F_{\nu\beta}+\frac{1}{4}\delta^{\alpha}_\nu F^{\gamma\delta}F_{\gamma\delta}.
\end{align}
So this was a happy example.
My questions are:


*

*What would happen if I can't do the 'improved translation' always?

*Energy-momentum tensor should always be gauge-invariant? What does imply this? (Furthermore, what would be the consequences at the quantum theory?)

*There's some examples of not gauge-invariant energy-momentum tensor? (or perhaps some loss of - some more (or less) "dramatic" - symmetry?)

 A: This answer was provided by OP as an edit to the question:


I will answer with a concrete example.
"Energy-momentum tensor should always be gauge-invariant? What does imply this? (Furthermore, what would be the consequences at the quantum theory?)". We want it to be gauge-invariant because energy and momentum will be a measure and it must be the ideal case, nevertheless, it'll not always be gauge-invariant and it's fine. Consider the gravitational wave theory, whose action is the linearised general relativity action. The energy-momentum tensor $t_{\mu\nu}$ will not be gauge invariant. As a consequence we will consider the mean value of it $\langle t_{\mu\nu}\rangle$ in order to obtain a dependence of the physical modes $h_{ij}^{TT}$ [1]. $\langle t_{\mu\nu}\rangle$ will be the observable indeed, and not $t_{\mu\nu}$.
"There's some examples of not gauge-invariant energy-momentum tensor? (or perhaps some loss of - some more (or less) "dramatic" - symmetry?)". Yup, the same example exposed in [1] for gravitational waves, which is a nice example because was measured in 2016.
[1] Maggiore, Michele - Gravitational Waves, Vol. 1: Theory and Experiments (equation (1.136))

A: I see no reason why a theory should be gauge invariant$^*$. It would certainly be interesting to see a list of reasons for this fundamental requirement.
Now to your question whether such a, in your words, 'particular gauge transformation' is always possible, The answer is yes. This trick can be written in terms of the Lagrangian without specifying it. One may still question if the result is really the energy-momentum density. The precise form of the energy-momentum distribution is relevant as it is a source of gravity.
$^*$ Please check out my non gauge invariant formulation of electromagnetism at https://arxiv.org/abs/physics/0106078 published as EJP D 8,p9-12(2000).
