  The first detailed study on the performance of a ZnO-based cryogenic
scintillating bolometer as a detector to search for rare processes in zinc
isotopes was performed. A 7.2 g ZnO low-temperature detector, containing more
than 80\% of zinc in its mass, exhibits good energy resolution of baseline
noise 1.0--2.7 keV FWHM at various working temperatures resulting in a
low-energy threshold for the experiment, 2.0--6.0 keV. The light yield for
$\beta$/$\gamma$ events was measured as 1.5(3) keV/MeV, while it varies for
$\alpha$ particles in the range of 0.2--3.0 keV/MeV. The detector demonstrate
an effective identification of the $\beta$/$\gamma$ events from $\alpha$ events
using time-properties of only heat signals. %(namely, Rise time parameter). The
radiopurity of the ZnO crystal was evaluated using the Inductively Coupled
Plasma Mass Spectrometry, an ultra-low-background High Purity Ge
$\gamma$-spectrometer, and bolometric measurements. Only limits were set at the
level of $\mathcal{O}$(1--100) mBq/kg on activities of \Nuc{K}{40},
\Nuc{Cs}{137} and daughter nuclides from the U/Th natural decay chains. The
total internal $\alpha$-activity was calculated to be 22(2) mBq/kg, with a
major contribution caused by 6(1) mBq/kg of \Nuc{Th}{232} and 12(2) mBq/kg of
\Nuc{U}{234}. Limits on double beta decay (DBD) processes in \Nuc{Zn}{64} and
\Nuc{Zn}{70} isotopes were set on the level of
$\mathcal{O}(10^{17}$--$10^{18})$ yr for various decay modes profiting from 271
h of acquired background data in the above-ground lab. This study shows a good
potential for ZnO-based scintillating bolometers to search for DBD processes of
Zn isotopes, especially in \Nuc{Zn}{64}, with the most prominent spectral
features at $\sim$10--20 keV, like the two neutrino double electron capture. A
10 kg-scale experiment can reach the experimental sensitivity at the level of
$\mathcal{O}(10^{24})$ yr.
