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Spatial curvature is one of the most fundamental parameters in our currentconcordance flat $\Lambda$CDM model of the Universe. The goal of this work isto investigate how the constraint on the spatial curvature is affected by anassumption on the sound horizon scale. The sound horizon is an essentialquantity to use the standard ruler from the Cosmic Microwave Background (CMB)and Baryon Acoustic Oscillations (BAOs). As an example, we study the curvatureconstraint in an axion-like Early Dark Energy (EDE) model in light of recentcosmological datasets from Planck, the South Pole Telescope (SPT), and theAtacama Cosmology Telescope (ACT), as well as BAO data compiled in SloanDigital Sky Survey Data Release 16. We find that, independent of the CMBdatasets, the EDE model parameters are constrained only by the CMB powerspectra as precisely and consistently as the flat case in previous work, evenwith the spatial curvature. We also demonstrate that combining CMB with BAO isextremely powerful to constrain the curvature parameter even with a reductionof the sound-horizon scale in an EDE model, resulting in $\Omega_K=-0.0056\pm0.0031$ in the case of ACT+BAO after marginalizing over the parameters of theEDE model. This constraint is as competitive as the Planck+BAO result in a$\Lambda$CDM model, $\Omega_{K}=-0.0001\pm 0.0018$.

Constraining the spatial curvature with cosmic expansion history in a cosmological model with a non-standard sound horizon