  Modified gravity theories are a popular alternative to dark energy as a
possible explanation for the observed accelerating cosmic expansion, and their
cosmological tests are currently an active research field. Studies in recent
years have been increasingly focused on testing these theories in the nonlinear
regime, which is computationally demanding. Here we show that, under certain
circumstances, a whole class of theories can be ruled out by using background
cosmology alone. This is possible because certain classes of models (i) are
fundamentally incapable of producing specific background expansion histories,
and (ii) said histories are incompatible with local gravity tests. As an
example, we demonstrate that a popular class of models, $f(R)$ gravity, would
not be viable if observations suggest even a slight deviation of the background
expansion history from that of the $\Lambda$CDM paradigm.
