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We study the viable Starobinsky <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>(</mo><mi>R</mi><mo>)</mo></mrow></semantics></math></inline-formula> dark energy model in spatially non-flat FLRW backgrounds, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow><mo>=</mo><mi>R</mi><mo>−</mo><mi>λ</mi><msub><mi>R</mi><mrow><mi>c</mi><mi>h</mi></mrow></msub><mrow><mo>[</mo><mn>1</mn><mo>−</mo><msup><mrow><mo>(</mo><mn>1</mn><mo>+</mo><msup><mi>R</mi><mn>2</mn></msup><mo>/</mo><msubsup><mi>R</mi><mrow><mi>c</mi><mi>h</mi></mrow><mn>2</mn></msubsup><mo>)</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>]</mo></mrow></mrow></semantics></math></inline-formula> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>R</mi><mrow><mi>c</mi><mi>h</mi></mrow></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>λ</mi></semantics></math></inline-formula> representing the characteristic curvature scale and model parameter, respectively. We modify CAMB and CosmoMC packages with the recent observational data to constrain Starobinsky <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>(</mo><mi>R</mi><mo>)</mo></mrow></semantics></math></inline-formula> gravity and the density parameter of curvature <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="sans-serif">Ω</mi><mi>K</mi></msub></semantics></math></inline-formula>. In particular, we find the model and density parameters to be <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>λ</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo><</mo><mn>0.283</mn></mrow></semantics></math></inline-formula> at 68% C.L. and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="sans-serif">Ω</mi><mi>K</mi></msub><mo>=</mo><mo>−</mo><mn>0</mn><mo>.</mo><msubsup><mn>00099</mn><mrow><mo>−</mo><mn>0.0042</mn></mrow><mrow><mo>+</mo><mn>0.0044</mn></mrow></msubsup></mrow></semantics></math></inline-formula> at 95% C.L., respectively. The best <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>χ</mi><mn>2</mn></msup></semantics></math></inline-formula> fitting result shows that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>χ</mi><mrow><mi>f</mi><mo>(</mo><mi>R</mi><mo>)</mo></mrow><mn>2</mn></msubsup><mo>≲</mo><msubsup><mi>χ</mi><mrow><mi mathvariant="sans-serif">Λ</mi><mi>C</mi><mi>D</mi><mi>M</mi></mrow><mn>2</mn></msubsup></mrow></semantics></math></inline-formula>, indicating that the viable <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>(</mo><mi>R</mi><mo>)</mo></mrow></semantics></math></inline-formula> gravity model is consistent with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="sans-serif">Λ</mi></semantics></math></inline-formula>CDM when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="sans-serif">Ω</mi><mi>K</mi></msub></semantics></math></inline-formula> is set as a free parameter. We also evaluate the values of AIC, BIC and DIC for the best fitting results of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>(</mo><mi>R</mi><mo>)</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="sans-serif">Λ</mi></semantics></math></inline-formula>CDM models in the non-flat universe.