$T^{00} ≠ $ Hamiltonian Density?

Check page 48-49 of http://walterpfeifer.ch/qft/QFT5.pdf?. It is apparent that the Hamiltonian density of the Maxwell Field is not positive definite when expressed in terms of the Four-vector Potential $A^mu$, even if the total Hamiltonian corresponds to $$int frac{1}{2} (vec{E}^2 +vec{B}^2)d^3x$$
Why is it that when the Gauge Potential is used, we end up with a Hamiltonian Density which isn’t equal to $frac{1}{2} (vec{E}^2 +vec{B}^2)$ ? Most importantly, does this mean that the Hamiltonian Density does not necessarily correspond to the $T^{00}$ energy-momentum tensor (since the energy momentum tensor must be gauge-invariant?)