Holographic entropy bounds are upper bounds for the maximum amount of entropy a given region can have, rather than its current entropy--for example, the spherical region of space just large enough to enclose the Earth and its atmosphere would currently contain a much lower entropy than a spherical black hole of the same radius, whose entropy is given by the Bekenstein bound. The scholarpedia article on the Bekenstein bound mentions the issue of cosmological entropy bounds:
Problems with bound (1) or the holographic bound are known to occur in extreme situations. Both fail when the gravitational potential is large (strong self-gravity), e.g. system already collapsed inside a black hole. In an infinite universe the holographic bound fails when applied to a sufficiently large region. In a closed (finite) universe the specification of R or bounding area becomes ambiguous (Bousso 1999, 2002). The covariant entropy bound corrects these and other shortcomings.
So apparently the "covariant entropy bound" (also known as Bousso's holographic bound) is the thing to look into when considering the maximum entropy of the universe (and I assume you just mean the observable universe here rather than whatever may lie beyond it). This set of lecture slides by Raphael Bousso, the physicist who originally proposed the covariant entropy bound, says that this bound is "conjectured to hold in arbitrary spacetimes, including cosmology." But it also says "If correct, origin must lie in quantum gravity", probably meaning one can't derive this fact in a non-conjectural way from basic laws of physics without having a complete theory of quantum gravity.
For an introduction to this stuff, I recommend this Scientific American article that discusses various holographic bonds, including the Bekenstein bound along with Bousso's conjectured more general bound (the article was written by Bekenstein himself). Though it is from 2003 so I'm not sure what kind of theoretical progress has been made since then. Anyway, here's what it says about Buosso's entropy bound:
In 1999 Raphael Bousso, then at Stanford, proposed a modified holographic bound, which has since been found to work even in situations where the bounds we discussed earlier cannot be applied. Bousso’s formulation starts with any suitable 2-D surface; it may be closed like a sphere or open like a sheet of paper. One then imagines a brief burst of light issuing simultaneously and perpendicularly from all over one side of the surface. The only demand is that the imaginary light rays are converging to start with. Light emitted from the inner surface of a spherical shell, for instance, satisfies that requirement. One then considers the entropy of the matter and radiation that these imaginary rays traverse, up to the points where they start crossing. Bousso conjectured that this entropy cannot exceed the entropy represented by the initial surface—one quarter of its area, measured in Planck areas. This is a different way of tallying up the entropy than that used in the original holographic bound.
Bousso’s bound refers not to the entropy of a region at one time but rather to the sum of entropies of locales at a variety of times: those that are “illuminated” by the light burst from the surface. Bousso’s bound subsumes other entropy bounds while avoiding their limitations. Both the universal entropy bound and the ’t Hooft-Susskind form of the holographic bound can be deduced from Bousso’s for any isolated system that is not evolving rapidly and whose gravitational field is not strong. When these conditions are overstepped—as for a collapsing sphere of matter already inside a black hole—these bounds eventually fail, whereas Bousso’s bound continues to hold. Bousso has also shown that his strategy can be used to locate the 2-D surfaces on which holograms of the world can be set up.
But again, this bound should be the upper limit on the entropy of a given region, not necessarily the actual entropy in that region, so this type of holographic bound shouldn't imply that when we increase the entropy of the universe, the universe needs to grow. This paper specifically discusses how most matter-containing regions in cosmology apart from black holes would fail to "saturate" the covariant entropy bound, meaning that the actual entropy in the region is smaller than the maximum possible according to the bound.