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By extending the standard holographic principle to a cosmological framework and combining the non-flat condition with the Kaniadakis entropy, we construct the non-flat Kaniadakis holographic dark energy model. The model employs Kaniadakis parameter $K$ and a parameter $c$. Derivation of the differential equation for KHDE density parameter to describe the evolutionary behavior of the universe is obtained. Such a differential equation could explain both the open as well as closed universe models. The classification based on matter and dark energy (DE) dominated regimes show that the KHDE scenario may be used to specify the Universe's thermal history and that a quintom regime can be encountered. For open and closed both the cases, we find the expressions for the deceleration parameter and the equation of state (EoS) parameter. Also, by varying the associated parameters, classical stability of the method is established. On considering the curvature to be positive, the universe favors the quintom behavior for substantially smaller values as opposed to the flat condition, when only quintessence is attained for such $K$ values. Additionally, we see a similar behavior while considering the curvature to be negative for such $K$ values. Therefore, adding a little bit of spatial geometry that isn't flat to the KHDE enhances the phenomenology while maintaining $K$ values at lower levels. To validate the model parameters, the most recent $30\;H(z)$ dataset measurements, in the redshift range $0.07 \leq z \leq 1.965$ are utilized. In addition, the distance modulus measurement from the current Union 2.1 data set of type Ia supernovae are employed.