Background and purpose: In online-adaptive proton therapy planning based on cone beam computed tomography (CBCT), CT number errors can pose challenges. We propose an approach for coping with CT number uncertainties by increasing range robustness settings (RRS) in online-adaptive planning. This was compared to our trigger-based offline (TB-Offline) adaptive approach, and to daily replanning using in-room CT-on-rails (CTOR). Material and methods: For 23 head-and-neck cancer patients, a CTOR and CBCT were acquired in a single fraction. CTOR contours were copied rigidly onto the CBCT. CBCT-based plans were generated with 3, 6, 8, 10, and 12 % RRS, each with 1 mm setup-RS, followed by a forward dose calculation on the reference CTOR. This was compared to dose distributions from our TB-Offline approach (3 mm/3% SRS/RRS), also recomputed on the CTOR. Coverage (voxelwise-minimum) of the primary clinical target volume (CTV₇₀₀₀) and elective lymph nodes (CTV₅₄₂₅) and grade ≥ II normal tissue complication probabilities were compared between strategies. Results: When going from RRS = 3 % to RRS = 10 %, the population 90th percentiles of CTV₅₄₂₅ V_94% improved from 89.6 % to 96.4 %, and CTV₇₀₀₀ V_94% from 92.8 % to 96.4 %. Substantial coverage loss (V_94%<95 %) with CBCT-based online adaptive and RRS = 10 % was observed in 1/23 evaluated patients for CTV₇₀₀₀ and 2/23 for CTV₅₄₂₅. This was an improvement compared to 3/23 and 4/23 with TB-Offline. Moreover, for RRS = 10 % the average risk of xerostomia improved by 2.4 percentage point compared to TB-Offline. Conclusions: Robust optimization with increased range robustness settings effectively mitigated dose degradation from CT number errors in CBCT-based online-adaptive proton therapy.