  In this work, we consider four $f(R)$ gravity models -- the Hu-Sawicki,
Starobinsky, Exponential and Tsujikawa models -- and use a range of
cosmological data, together with Markov Chain Monte Carlo sampling techniques,
to constrain the associated model parameters. Our main aim is to compare the
results we get when $\Omega_{k,0}$ is treated as a free parameter with their
counterparts in a spatially flat scenario. The bounds we obtain for
$\Omega_{k,0}$ in the former case are compatible with a flat geometry. It
appears, however, that a higher value of the Hubble constant $H_0$ allows for
more curvature. Indeed, upon including in our analysis a Gaussian likelihood
constructed from the local measurement of $H_0$, we find that the results favor
an open universe at a little over $1\sigma$. This is perhaps not statistically
significant, but it underlines the important implications of the Hubble tension
for the assumptions commonly made about spatial curvature. We note that the
late-time deviation of the Hubble parameter from its $\Lambda$CDM equivalent is
comparable across all four models, especially in the non-flat case. When
$\Omega_{k,0}=0$, the Hu-Sawicki model admits a smaller mean value for
$\Omega_{\text{cdm},0}h^2$, which increases the said deviation at redshifts
higher than unity. We also study the effect of a change in scale by evaluating
the growth rate at two different wavenumbers $k_\dagger$. Any changes are, on
the whole, negligible, although a smaller $k_\dagger$ does result in a slightly
larger average value for the deviation parameter $b$.
