The bandgap of a semiconductor, in general, should decrease with temperature. Here are a few papers on the subject:
[1] W. Bludau, A. Onton, and W. Heinke, Temperature Dependence of the Band Gap of Silicon, Journal of Applied Physics, 45, 1846 (1974), 10.1063/1.1663501.
[2] Y. P. Varshni, Temperature Dependence of the Energy Gap in Semiconductors, Physica, 34, 149 (1967), 10.1016/0031-8914(67)90062-6.
[3] L. Viña, S. Logothetidis, and M. Cardona, Temperature Dependence of the Dielectric Function of Germanium, Physical Review B, 30, 1979 (1984), 10.1103/physrevb.30.1979.
[4] V. Alex, S. Finkbeiner, and J. Weber, Temperature Dependence of the Indirect Energy Gap in Crystalline Silicon, Journal of Applied Physics, 79, 6943 (1996), 10.1063/1.362447.
[5] P. Lautenschlager, M. Garriga, L. Vina, and M. Cardona, Temperature Dependence of the Dielectric Function and Interband Critical Points in Silicon, Physical Review B, 36, 4821 (1987), 10.1103/physrevb.36.4821.
[6] N. M. Ravindra and V. K. Srivastava, Temperature Dependence of the Energy Gap in Semiconductors, Journal of Physics and Chemistry of Solids, 40, 791 (1979), 10.1016/0022-3697(79)90162-8.
[7] K. P. O’Donnell and X. Chen, Temperature Dependence of Semiconductor Band Gaps, Applied Physics Letters, 58, 2924 (1991), 10.1063/1.104723.
[8] R. Passler, Parameter Sets Due to Fittings of the Temperature Dependencies of Fundamental Bandgaps in Semiconductors, Physica Status Solidi (B), 216, 975 (1999), 10.1002/(sici)1521-3951(199912)216:23.0.co;2-n.
[9] P. Lautenschlager, M. Garriga, S. Logothetidis, and M. Cardona, Interband Critical Points of GaAs and Their Temperature Dependence, Physical Review B, 35, 9174 (1987), 10.1103/physrevb.35.9174.
One of the main reasons for the change in the bandgap size is due to the electron-phonon coupling. It should be noted, however, that the bandgap of each material might have its own temperature dependence, and in some particular cases the bandgap might increase with temperature.
If the bandgap of a material composing a solar cell increases, I believe its efficiency could drop, but I'm not an expert in that field, I only know the very basic about solar cells. However, I do know that the best material for solar cell applications would be silicon due to its bandgap that is optimal for light absorption while maintaining a low scattering rate.
An increase in temperature will always affect the semiconductor properties (semiconductors are sensitive to temperature) i.e., reduce the bandgap of the semiconductor where the energy needed for the excitation of electrons will be smaller, which results in reduced open circuit voltage (Voc) and efficiency of solar cells. The solar cell's power output directly depends on quasi-Fermi level separation, i.e., Voc. Though, the decrease in bandgap extends the absorption range and increases the short-circuit current (Jsc) of the solar cell, but also decreases the open-circuit voltage (Voc) of the device, thus a half-and-half situation arises for the optimization of the semiconductor material. So Normally, the standard test condition of solar cells in the laboratory is done at 25 degrees Celsius to avoid temperature effects.
You can also refer this article,
Simultaneously Decreasing the Bandgap and Voc Loss in Efficient Ternary Organic Solar Cells.