This study investigates the nonlinear thermomechanical buckling and postbuckling behavior of porous sandwich cylindrical shells. The shells consist of a graphene-origami-reinforced auxetic metamaterial core and porous carbon nanotube (CNT)-reinforced face sheets, supported by a Winkler–Pasternak elastic foundation and subjected to external pressure in a thermal environment. To assess the influence of porosity, various porosity distribution models are examined across the shell thickness. Governing equations are formulated based on Reddy's third-order shear deformation theory (TSDT) and solved via the Galerkin method. Validation against established solutions in the literature demonstrates the high accuracy and reliability of the proposed model. A comprehensive parametric study is conducted to examine the effects of porosity characteristics, temperature variations, auxetic core, and Winkler Pasternak foundation properties on the buckling and postbuckling responses. The findings reveal that both porosity and temperature significantly influence the nonlinear buckling behavior, underscoring the necessity of accounting for these effects in designing advanced auxetic sandwich structures.