  Deuterated molecules are good tracers of the evolutionary stage of
star-forming cores. During the star formation process, deuterated molecules are
expected to be enhanced in cold, dense pre-stellar cores and to deplete after
protostellar birth. In this paper we study the deuteration fraction of
formaldehyde in high-mass star-forming cores at different evolutionary stages
to investigate whether the deuteration fraction of formaldehyde can be used as
an evolutionary tracer. Using the APEX SEPIA Band 5 receiver, we extended our
pilot study of the $J$=3$\rightarrow$2 rotational lines of HDCO and D$_2$CO to
eleven high-mass star-forming regions that host objects at different
evolutionary stages. High-resolution follow-up observations of eight objects in
ALMA Band 6 were performed to reveal the size of the H$_2$CO emission and to
give an estimate of the deuteration fractions HDCO/H$_2$CO and D$_2$CO/HDCO at
scales of $\sim$6" (0.04-0.15 pc at the distance of our targets). Our
observations show that singly- and doubly deuterated H$_2$CO are detected
toward high-mass protostellar objects (HMPOs) and ultracompact HII regions
(UCHII regions), the deuteration fraction of H$_2$CO is also found to decrease
by an order of magnitude from the earlier HMPO phases to the latest
evolutionary stage (UCHII), from $\sim$0.13 to $\sim$0.01. We have not detected
HDCO and D$_2$CO emission from the youngest sources (high-mass starless cores,
HMSCs). Our extended study supports the results of the previous pilot study:
the deuteration fraction of formaldehyde decreases with evolutionary stage, but
higher sensitivity observations are needed to provide more stringent
constraints on the D/H ratio during the HMSC phase. The calculated upper limits
for the HMSC sources are high, so the trend between HMSC and HMPO phases cannot
be constrained.
