We consider the thermodynamic and stability problem of Kerr black holes arising from the nonextensive/nonadditive nature of the Bekenstein-Hawking entropy formula. Nonadditive thermodynamics is often criticized by asserting that the zeroth law cannot be compatible with nonadditive composition rules, so in this work we follow the so-called formal logarithm method to derive an additive entropy function for Kerr black holes satisfying also the zeroth law's requirement. Starting from the most general, equilibrium compatible, nonadditive entropy composition rule of Abe, we consider the simplest, non-parametric approach that is generated by the explicit nonadditive form of the Bekenstein-Hawking formula. This analysis extends our previous results on the Schwarzschild case and shows that the zeroth law compatible temperature function in the model is independent of the mass-energy parameter of the black hole. By applying the Poincaré turning point method we also study the thermodynamic stability problem in the system.