  Spatial curvature is one of the most fundamental parameters in our current
concordance flat $\Lambda$CDM model of the Universe. The goal of this work is
to investigate how the constraint on the spatial curvature is affected by an
assumption on the sound horizon scale. The sound horizon is an essential
quantity to use the standard ruler from the Cosmic Microwave Background (CMB)
and Baryon Acoustic Oscillations (BAOs). As an example, we study the curvature
constraint in an axion-like Early Dark Energy (EDE) model in light of recent
cosmological datasets from Planck, the South Pole Telescope (SPT), and the
Atacama Cosmology Telescope (ACT), as well as BAO data compiled in Sloan
Digital Sky Survey Data Release 16. We find that, independent of the CMB
datasets, the EDE model parameters are constrained only by the CMB power
spectra as precisely and consistently as the flat case in previous work, even
with the spatial curvature. We also demonstrate that combining CMB with BAO is
extremely powerful to constrain the curvature parameter even with a reduction
of the sound-horizon scale in an EDE model, resulting in $\Omega_K=-0.0056\pm
0.0031$ in the case of ACT+BAO after marginalizing over the parameters of the
EDE model. This constraint is as competitive as the Planck+BAO result in a
$\Lambda$CDM model, $\Omega_{K}=-0.0001\pm 0.0018$.
