Observational studies clearly reveal that natural space plasmas generally possess a pronounced non-Maxwellian high-energy tail distribution that can be well modeled by a generalized Lorentzian (kappa) distribution. In this study we consider the whistler mode wave instability driven by the anisotropy condition (T$perp$/T$parallel$ > 1) of energetic electrons modeled with a typical kappa distribution in the presence of a cold plasma population. We use a linear theory to study the instability threshold condition for two typical plasma regions of interest: the higher-density (or a weakly magnetized) region and the lower-density (or a strongly magnetized) region. We find that (1) as in the case for a regular bi-Maxwellian, the energetic electron anisotropy T$perp$/T$parallel$ is subject to the threshold condition of this whistler instability, and the instability threshold condition obeys a general form T$perp$/T$parallel$ - 1 = S/¿$parallel$α, with a narrow range of the fitting parameter 0.25 ≤ α ≤ 0.52 over 0.01 ≤ ¿$parallel$ ≤ 2.0; (2) the instability threshold condition in the higher-density (or a weakly magnetized) region is generally lower than that in the lower-density (or a strongly magnetized) region, specifically, with the fitting parameter range 0.3 ≤ S ≤ 5.0 in the higher-density (or a weakly magnetized) region, while 0.32 ≤ S ≤ 6.94 in the lower-density (or a strongly magnetized) region; and (3) the instability threshold condition for the kappa distribution generally decreases as the spectral index $kappa$ increases and tends to the lowest limiting values of the bi-Maxwellian as $kappa$ → ∞. The results above may present a further insight into the nature of this instability threshold condition for the whistler mode waves in the outer radiation belts of the Earth, the inner Jovian magnetosphere, or other space plasmas where an anisotropic hot electron component and a cold plasma component are both present.