  We explore the viable $f(R)$ gravity models in FLRW backgrounds with a free
spatial curvature parameter $\Omega_{K}$. In our numerical calculation, we
concentrate on the exponential $f(R)$ model of $f(R) = R - \lambda
R_{ch}(1-\exp{(-R/R_{ch}}))$, where $R_{ch}$ is the characteristic curvature
scale, which is independent of $\Omega_K$, and $\lambda$ corresponds to the
model parameter, while $R_{ch}\lambda=2\Lambda$ with $\Lambda$ the cosmological
constant. In particular, we study the evolutions of the dark energy density and
equation of state for exponential $f(R)$ gravity in open, flat and closed
universe, and compare with those for $\Lambda$CDM. From the current
observational data, we find that $\lambda^{-1}=0.42927^{+0.39921}_{-0.32927}$
at 68$\%$ C.L and $\Omega_K=-0.00050^{+0.00420}_{-0.00414}$ at 95$\%$ C.L. in
the exponential $f(R)$ model. By using Akaike information criterion (AIC),
Bayesian information criterion (BIC) and Deviance Information Criterion (DIC),
we conclude that there is no strong preference between the exponential $f(R)$
gravity and $\Lambda$CDM models in the non-flat universe.
