  We construct a stable right inverse for the divergence operator in
non-cylindrical domains in space-time. The domains are assumed to be H\"older
regular in space and evolve continuously in time. The inverse operator is of
Bogovskij type, meaning that it attains zero boundary values. We provide
estimates in Sobolev spaces of positive and negative order with respect to both
time and space variables. The regularity estimates on the operator depend on
the assumed H\"older regularity of the domain. The results can naturally be
connected to the known theory for Lipschitz domains. As an application, we
prove refined pressure estimates for weak and very weak solutions to
Navier--Stokes equations in time dependent domains.
