One of the features of the unconventional $s_pm$ state in iron-based superconductors is possibility to transform to the $s_{++}$ state with the increase of the nonmagnetic disorder. Detection of such a transition would prove the existence of the $s_pm$ state. Here we study the temperature dependence of the London magnetic penetration depth within the two-band model for the $s_pm$ and $s_{++}$ superconductors. By solving Eliashberg equations accounting for the spin-fluctuation mediated pairing and nonmagnetic impurities in the $T$-matrix approximation, we have derived a set of specific signatures of the $s_pm to s_{++}$ transition: (1) sharp change in the behavior of the penetration depth $lambda_{L}$ as a function of the impurity scattering rate at low temperatures; (2) before the transition, the slope of $Delta lambda_{L}(T) = lambda_{L}(T)-lambda_{L}(0)$ increases as a function of temperature, and after the transition this value decreases; (3) the sharp jump in the inverse square of the penetration depth as a function of the impurity scattering rate, $lambda_{L}^{-2}(Gamma_a)$, at the transition; (4) change from the single-gap behavior in the vicinity of the transition to the two-gap behavior upon increase of the impurity scattering rate in the superfluid density $rho_{s}(T)$.
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