  Loop quantum cosmology (LQC) seems to be predicting modified effective
Friedmann equations without extra degrees of freedom. A puzzle arises if one
decides to seek for a covariant effective action which would lead to the given
Friedmann equation: The Einstein--Hilbert action is the only action that leads
to second order field equations and, hence, there exists no covariant action
which, under metric variation, leads to modified Friedmann equation without
extra degrees of freedom. It is shown that, at least for isotropic models in
LQC, this issue is naturally resolved and a covariant effective action can be
found if one considers higher order theories of gravity but faithfully follows
effective field theory techniques. However, our analysis also raises doubts on
whether a covariant description without background structures can be found for
anisotropic models.
