The question you should be asking yourself is: what properties do I want to compute? You, naturally, want to make sure that property is converged. For instance, if you wish to compute the band gap, then it makes more sense to confirm that your band gap is converged with your given settings. The band gap could very well be more (or less) sensitive to a given parameter than the absolute energy, in which case your convergence test on the absolute energy may not translate well for the property of interest. There is no inherent reason to do a convergence test on the absolute energy -- it is just a commonly computed property and generally will indicate that other properties are well-converged, but that's certainly not a guarantee.
As an aside, there may be cases where converging the absolute energy is recommended but not required. If you are computing a reaction energy or any other energy for that matter, what you are often reporting is an energy difference. In many cases, there is significant cancellation of error that occurs, which means that the energy difference will often converge faster than the absolute energy.
Interestingly enough, there are even subtle cases with certain codes where it is impossible to converge the absolute energy. I found this to be the case in VASP when using the
LASPH flag, which determines whether to include non-spherical contributions related to the gradient of the density in the projector-augmented wave spheres. When enabling
LASPH, the absolute energy never converges with respect to increasing plane-wave kinetic energy cutoff because the non-spherical contribution is always changing. In this case, the only option is to check that an energy difference is converged.