The particle velocity distribution in space plasma usually exhibits a non-Maxwellian high-energy tail that can be well modeled by kappa distributions. In this study, we focus on the growth rates of the Alfvén-cyclotron instability driven by ion temperature anisotropy in a kappa plasma. By solving the kinetic linear dispersion equation, we explore the sensitivity of growth rates to the spectral index κ of a bi-kappa distribution under different plasma conditions, including a variety of plasma beta \beta _hp and temperature anisotropy A_hp values of hot protons. Furthermore, a concise, analytic scaling formula is derived that relates the dimensionless maximum growth rate to three independent variables: the spectral index and the plasma beta and temperature anisotropy of hot protons. Our results show that as the κ-value increases, the instability bandwidth narrows and the maximum growth rate increases significantly. For higher \beta _hp and A_hp , the maximum instability undergoes a sharp increase as well. When our fits of dimensionless maximum growth rates are compared with solutions to kinetic linear dispersion theory, the results generally exhibit good agreement between them. Especially under the circumstances of large κ-values and high \beta _hp and A_hp , the scalings of maximum growth rates primarily accurately model the numerical solutions. Our analytic expressions can readily be used in large-scale models of the Earth’s magnetosphere to understand wave generation due to the Alfvén-cyclotron instability.