Anisotropic charge-carrier transport in black phosphorus limited by ionized impurity scattering at finite temperature is explored theoretically. The anisotropic electronic structure enters the calculation for the polarizability (screening), the momentum relaxation time, and the mobility. For finite temperature, elastic scattering is not limited to the Fermi surface and the polarizability is temperature dependent. The impact of screening is investigated in detail with its dependence on carrier density and temperature. Competing with the thermal excitation effects, the temperature dependence of the polarizability is found to dominate for T<100 K. As a result, the charge-carrier mobility slowly decreases with increasing temperature. The weak temperature dependence of the mobility and its anisotropy ratio of 1.9-3.2 agree with published experimental data.