In this paper, convex modeling based topology optimization with load uncertainty is presented. The load uncertainty is described using the non-probabilistic based unknown-but-bounded convex model, and the strain energy based topology optimization problem under uncertain loads is formulated. Unlike the conventional deterministic topology optimization problem, the maximum possible strain energy under uncertain loads is selected as the new objective in order to achieve a safe solution. Instead of obtaining approximated solutions as used before, an exact solution procedure is presented. The problem is first formulated as a single level optimization problem, and then rewritten as a two-level optimization problem. The upper level optimization problem is solved as a deterministic topology optimization with the load which generated from the worst structure response in the lower level problem. The lower level optimization problem is to identify this worst structure response, and it is found equivalent to an inhomogeneous eigenvalue problem. Three different cases are discussed for accurately evaluating the global optima of the lower level optimization problem, while the corresponding sensitivities are derived individually. With the function value and sensitivity information ready, the upper level optimization problem can be solved through existing gradient based optimization algorithms. The effectiveness of the proposed convex modeling based topology optimization is demonstrated through different numerical examples.