I will try to answer in the view of General Relativity. 

I quote directly from Einstein and Rosen paper :

> The four-dimensional space is described mathematically by two
congruent parts or "sheets", corresponding to $u > 0$ and $u < 0$,
which are joined by a hyperplane $r = 2m$ or $u = 0$ in which $g$ vanishes. 
We call such a connection between the two sheets a "bridge".
We see now in the given solution, free from singularities, 
the mathematical representation of an elementary particle
(neutron or neutrino). Characteristic of the theory we are presenting is
the description of space by means of two sheets.
A bridge, spatially finite, which connects these sheets 
characterizes the presence of an electrically neutral elementary particle.
With this conception one not only obtains the representation of an
elementary particle by using only the field equations, that is, without 
introducing new field quantities to describe the density of matter; one is also
able to understand the atomistic character of matter **as well as the fact that
there can be no particles of negative mass**. The latter is made clear
by the following considerations. If we had started from a Schwarzschild 
solution with negative $m$, we should not have been able to make the 
solution regular by introducing a new variable $u$ instead of $r$; 
that is to say, *no "bridge" is possible that corresponds to a particle of negative mass*.

[A.Einstein, N.Rosen - "The Particle Problem in the General Theory of Relativity"][1]

This was the paper that investigated ER bridges (wormholes). It would be useful to read the paper to understand further more *why there can be no negative mass particles*.

So, **the total energy of the universe can not be zero**, because there can be no particles of negative mass. So, in the light of mass/energy equivalence, the energy of the universe cannot be zero.


  [1]: https://journals.aps.org/pr/abstract/10.1103/PhysRev.48.73