  We examine the third quantization of $f(R)$-type gravity, based on its
effective Lagrangian in the case of a flat Friedmann-Lemaitre-Robertson-Walker
metric. Starting from the effective Lagrangian, we execute a suitable change of
variable and the second quantization, and we obtain the Wheeler-DeWitt
equation. The third quantization of this theory is considered. And the
uncertainty relation of the universe is investigated in the example of
$f(R)$-type gravity, where $f(R)=R^2$. It is shown, when the time is late
namely the scale factor of the universe is large, the spacetime does not
contradict to become classical, and, when the time is early namely the scale
factor of the universe is small, the quantum effects are dominating.
