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In this work, we measure two important phenomenological parameters of corona(and hot accretion flow) in black hole X-ray binaries: the photon index$\Gamma$ and the electron temperature $\te$. Thanks to the capability of {\itNuSTAR} in hard X-rays, we measure these two parameters over six orders ofmagnitude in the 0.1-100 keV X-ray luminosity $\lx$, from $\sim 5\times10^{38}\ergs$ down to as low as $\sim 5\times10^{32} \ergs$. We confirm the existenceof a "V"-shaped correlation between $\Gamma$ and $\lx$. Surprisingly, weobserve a "$\Lambda$"-shaped correlation between $\te$ and $\lx$. The "coolerwhen brighter" branch in the high luminosity regime ($\lx \ga 3\times 10^{36}\ergs$) agrees with previous results and can be understood under the existingmodel of Compton scattering in the corona. On the other hand, the apparent"cooler when fainter" (positive $\te$-$\lx$ correlation) branch in thelow-luminosity regime ($\lx \la 3\times 10^{36} \ergs$) is unexpected, thus itputs a new challenge to existing models of hot accretion flow/corona.

Coronal properties of black hole X-ray binaries in the hard state as seen by NuSTAR and Swift