Graphene is an atomically thin material with unique electrical, optical, and mechanical properties. In this thesis, we explore some of the interesting temperature-dependent mechanics of graphene membranes. We start by presenting the typical mechanical theory used by experimentalists to model a suspended graphene membrane in the presence of an electrostatic force, and we expand it to account for various effects, such as slack, capacitive softening, and dynamic changes in tension. We also perform finite element analysis using COMSOL Multiphysics software and compare the results with the analytic solution. Then, we show how to use the transfer matrix technique to model graphene optically as an infinitesimal conducting boundary. We solve for the reflectance of a graphene sheet parallel to a perfect mirror, which is important for measurements using optical detection. Next, we summarize the first measurement of photothermal optomechanics in graphene resonators, demonstrate both self-oscillation and cooling, and develop a theory to predict the optomechanical spring constant induced by photothermal forces. Finally, we develop an optical technique for sensing the static deflection of a graphene membrane and use it to measure the temperature dependence of the Young's modulus of graphene for the first time. We find that the room temperature modulus is much softer than expected from thermal rippling theories, but it stiffens significantly at low temperature.