Metal halide perovskites have proven to be excellent light-harvesting materials in photovoltaic devices whose efficiencies are rapidly improving. Here, we examine the temperature-dependent photon absorption, exciton binding energy, and band gap of FAPbI3 (thin film) and find remarkably different behavior across the β-γ phase transition compared with MAPbI3. While MAPbI3 has shown abrupt changes in the band gap and exciton binding energy, values for FAPbI3 vary smoothly over a range of 100-160 K in accordance with a more gradual transition. In addition, we find that the charge-carrier mobility in FAPbI3 exhibits a clear T-0.5 trend with temperature, in excellent agreement with theoretical predictions that assume electron-phonon interactions to be governed by the Fröhlich mechanism but in contrast to the T-1.5 dependence previously observed for MAPbI3. Finally, we directly observe intraexcitonic transitions in FAPbI3 at low temperature, from which we determine a low exciton binding energy of only 5.3 meV at 10 K.