Stability of motion and thermodynamics in charged black holes in $f(T)$ gravity

We investigate the stability of motion and the thermodynamics in the case of spherically symmetric solutions in $f(T)$ gravity using the perturbative approach. We consider small deviations from general relativity and we extract charged black hole solutions for two charge profiles, namely with or without a perturbative correction in the charge distribution. We examine their asymptotic behavior, we extract various torsional and curvature invariants, and we calculate the energy and the mass of the solutions. Furthermore, we study the stability of motion around the obtained solutions, by analyzing the geodesic deviation, and we extract the unstable regimes in the parameter space. We calculate the inner (Cauchy) and outer (event) horizons, showing that for larger deviations from general relativity or larger charges, the horizon disappears and the central singularity becomes a naked one. Additionally, we perform a detailed thermodynamic analysis examining the temperature, entropy, heat capacity and Gibb's free energy. Concerning the heat capacity we find that for larger deviations from general relativity it is always positive, and this shows that $f(T)$ modifications improve the thermodynamic stability, which is not the case in other classes of modified gravity.