We study the one-loop gluon polarization tensor at zero and finite temperature in the presence of a magnetic field, to extract the thermo-magnetic evolution of the strong coupling $alpha_s$. We analyze four distinct regimes, to wit, the small and large field cases, both at zero and at high temperature. From a renormalization group analysis we show that at zero temperature, either for small or large magnetic fields, and for a fixed transferred momentum $Q^2$, $alpha_s$ grows with the field strength with respect to its vacuum value. However, at high temperature and also for a fixed value of $Q^2$ we find two different cases: When the magnetic field is even larger than the squared temperature, $alpha_s$ also grows with the field strength. On the contrary, when the squared temperature is larger than the magnetic field, a turnover behavior occurs and $alpha_s$ decreases with the field strength. This thermo-magnetic behavior of $alpha_s$ can help explain the inverse magnetic catalysis phenomenon found by lattice QCD calculations.
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