According to Clinton Community College, temperature primarily affects the fluidity of a cell membrane, with temperatures that are too low causing it to solidify and temperatures that are too high causing it to become more fluid or even break up. Cell membranes at functional temperatures are relatively viscous liquid bubbles. Their structure is maintained by using a phospholipid bilayer and its inherent hydrophilic and hydrophobic qualities.
The phospholipid molecules of cell membranes are long molecules with very different groups at each end. At one end is a polar group of molecules that is attracted to water because of water's polar nature. At the other end are hydrocarbon chains which are non-polar and thus have no affinity for water. Both the cytoplasm of a cell and its immediate environment are full of water, so the polar sides naturally orient to face it. The two layers of phospholipids in a membrane allow the polar ends of the molecules to face water on both sides, while the non-polar ends are sheltered from the water between them. This creates a relatively stable and surprisingly rigid structure despite being liquid.
Like any fat or oil, the phospholipids in the cell membrane are more or less solid based on temperature. The membranes of cells in hot environments tend to use saturated fatty acid chains to stay more viscous, while those in cold environments use unsaturated chains to stay liquid.