We investigate soft, temperature-sensitive microgels at fluid interfaces. Though having an isotropic, spherical shape in bulk solution, the microgels become anisotropic upon adsorption. The structure of microgels at interfaces is described by a core-corona morphology. Here, we investigate how changing temperature across the microgel volume phase transition temperature, which leads to swelling/deswelling of the microgels in the aqueous phase, affects the phase behavior within the monolayer. We combine compression isotherms, atomic force microscopy imaging, multiwavelength ellipsometry, and computer simulations. At low compression, the interaction between adsorbed microgels is dominated by their highly stretched corona and the phase behavior of the microgel monolayers is the same. The polymer segments within the interface lose their temperature-sensitivity because of the strong adsorption to the interface. At high compression, however, the portions of the microgels that are located in the aqueous side of the interface become relevant and prevail in the microgel interactions. These portions are able to collapse and, consequently, the isostructural phase transition is altered. Thus, the temperature-dependent swelling perpendicular to the interface ("3D") affects the compressibility parallel to the interface ("2D"). Our results highlight the distinctly different behavior of soft, stimuli-sensitive microgels as compared to rigid nanoparticles.