  In this paper, we study cohomology theories of $\mathbb{Q}$-modulus pairs,
which are pairs $(X, D)$ consisting of a scheme $X$ and a $\mathbb{Q}$-divisor
$D$. Our main theorem provides a sufficient condition for such a cohomology
theory to be invariant under blow-ups with centers contained in the divisor.
This yields a short proof of the blow-up invariance of the Hodge cohomology
with modulus proved by Kelly-Miyazaki. We also define the Witt vector
cohomology with modulus using the Brylinski-Kato filtration and prove its
blow-up invariance.
