In one hospital ward it's +40°C, in another +10°C, and on average it's +25°C. If temperature affected something nonlinearly (say, the rate of bacterial growth) the average would give a wrong prediction. The same problem arises when cosmologists describe a universe where galaxies are packed into clusters and voids yawn between them.
Most calculations of cosmic expansion use 'uniform dust'—matter evenly smeared across space. It's convenient, but reality is different: matter is clumped. In standard gravity (general relativity), this substitution usually works because the coupling between gravity and matter is linear. But if gravity depends on matter nonlinearly (as in some extended theories, for instance when a function of the trace of the energy–momentum tensor is added to the equations) then averaging is no longer harmless.
The authors took a simple model — global monopoles, hypothetical point-like defects in space that could have formed in the early universe. They showed that if you average a nonlinear function of density over a volume and then plug in the average density, the result won't match what you get by first plugging in the real matter distribution and then averaging. The difference depends on how many particles are in a unit volume. For dust (ordinary pressureless matter), such theories give rise to a nonzero pressure, even though in standard cosmology dust has no pressure.
This means that in models with nonstandard gravity–matter coupling, you cannot ignore the universe's inhomogeneity. Otherwise, predictions about how fast the cosmos expands will be wrong. The universe is not just 'on average' homogeneous: its lumpiness can affect the most global processes.
Global monopoles are hypothetical objects that could have formed during phase transitions in the early universe, much like defects in a crystal lattice when water freezes. They haven't been detected yet, but they serve as a handy 'toy' for testing theories of gravity.
In Greg Egan's novel Schild's Ladder, physicists confront the possibility that the vacuum might be unstable due to exotic fields — a similar concern echoes in theories with nonstandard gravity, where 'empty' space might behave unexpectedly.