  The hard X-ray spectral index of some AGN has been observed to steepen with
the source flux. This has been interpreted in a Comptonization scenario, where
an increase in the soft flux decreases the temperature of the corona, leading
to steepening of the photon index. However, the variation of the coronal
temperature with flux has been difficult to measure due to the presence of
complex reflection component in the hard X-rays and the lack of high-quality
data at that energy band. Recently, a 200 ks {\it NuSTAR} observation of Ark
564 in 3--50 keV band, revealed the presence of one of the coolest coronae with
temperature $kT_e \sim 15$ keV in the time-averaged spectrum. Here, we
re-analyse the data and examined the spectra in four flux levels. Our analysis
shows that the coronal temperature decreased from $\sim 17$ to $\sim 14$ keV as
the flux increased. The high energy photon index $\Gamma \sim 2.3$ varied by
less than $0.1$, implying that the optical depth of the corona increased by
about 10\% as the flux increased. This first reporting of coronal temperature
variation with flux shows that further long observation by {\it NuSTAR} of this
and other sources would shed light on the geometry and dynamics of the inner
regions of the accretion flow.

High Energy Astrophysical Phenomena