  Barr--Beck cohomology, put into the framework of model categories by Quillen,
provides a cohomology theory for any algebraic structure, for example
Andr\'e--Quillen cohomology of commutative rings. Quillen cohomology has been
studied notably for divided power algebras and restricted Lie algebras, both of
which are instances of divided power algebras over an operad $P$: the
commutative and Lie operad respectively. In this paper, we investigate the
Quillen cohomology of divided power algebras over an operad $P$, identifying
Beck modules, derivations, and K\"ahler differentials in that setup. We also
compare the cohomology of divided power algebras over $P$ with that of
$P$-algebras, and work out some examples.
