In this work, we employ MFBO to search a pool of ∼600 covalent organic framework (COF) crystal structures for the one with the highest simulated xenon/krypton selectivity at room temperature, while incurring the minimal computational expense. We are armed with two molecular simulation methods to predict the Xe/Kr selectivity of a COF: (higher-fidelity & -cost) Markovchain Monte Carlo simulation of the binary grandcanonical ensemble, where the COF hosts multiple adsorbates (both Xe and Kr) during the simulation; and, (lower-fidelity & -cost) Monte Carlo integration to calculate the Xe and Kr Henry coefficients in the COF, which makes the dilute approximation, so the COF hosts only a single adsorbate during the simulation. Our MFBO routine employs (i) a multi-fidelity Gaussian process (GP) 64 surrogate model to predict the simulated Xe/Kr selectivity of a COF from its structural and chemical features and (ii) a cost-aware, multi-fidelity expected improvement acquisition function to design the next simulation. MFBO acquires the COF with the largest high-fidelity simulated Xe/Kr selectivity using only 38 low-and nine high-fidelity simulations, incurring only 2.5%, 5% on average, and 18% on average of the computational run time of a high-fidelity exhaustive search, random search, and single-fidelity BO, respectively. More, MFBO robustly out-performs single-fidelity BO, over randomly chosen COFs used to initialize the surrogate model. Our results demonstrate the promise of MFBO to cost-effectively discover materials for a variety of applications when in possession of multiple options of laboratory experiments and/or computer simulations, that trade fidelity for affordability, to measure/evaluate/predict the property of materials.