  [Abridged] Active Galactic Nuclei (AGN) are thought to be responsible for the
suppression of star formation in massive ~10$^{10}$ M$_\odot$ galaxies. While
this process is a key feature in numerical simulations, it is not yet
unambiguously confirmed in observational studies. Characterization of the star
formation rate (SFR) in AGN host galaxies is challenging as AGN light
contaminates most SFR tracers. We aim to obtain and compare SFR estimates from
different tracers for AGN host galaxies in the Close AGN Reference Survey
(CARS) to provide new observational insights. We construct integrated
panchromatic spectral energy distributions (SED) to measure the FIR luminosity
as a tracer for the recent (< 100 Myr) SFR. In addition, we use integral-field
unit observation of the CARS targets to employ the H$\alpha$ luminosity
decontaminated by AGN excitation as a proxy for the current (< 5 Myr) SFR. We
find that significant differences in specific SFR of the AGN host galaxies as
compared with the larger galaxy population disappear once cold gas mass, in
addition to stellar mass, is used to predict the SFR. We identify individual
galaxies with a significant difference in their SFR which can be related to a
recent enhancement or decline in their SFR history that might be related to
various processes including interactions, gas consumption, outflows and AGN
feedback. AGN can occur in various stages of galaxy evolution which makes it
difficult to relate the SFR solely to the impact of the AGN. We do not find any
strong evidence for global positive or negative AGN feedback in the CARS
sample. However, there is tentative evidence that 1) the relative orientation
of the AGN engine with respect to the host galaxies might alter the efficiency
of AGN feedback and 2) the recent SFH is an additional tool to identify rapid
changes in galaxy growth driven by the AGN or other processes.
