Aims. This paper presents experimentally obtained rate coefficients for the weakly endothermic reaction OD⁻ + H₂ → OH⁻ + HD with ortho- and para-hydrogen at astrophysically relevant temperatures between 10 and 300 K. Methods. The reaction was studied with normal and para-enriched (99.5% para-H₂) hydrogen in a 22-pole ion trap. The measured temperature dependencies of reaction rate coefficients are analyzed using a model which assumes that the rotational energies of the two reactants are equivalent to the translational energy in driving the reaction. Results. At room temperature, the rate coefficients of reactions with both nuclear spin variants reach 7 × 10⁻¹¹ cm³ s⁻¹, which is in good agreement with the previous results from ion trap and swarm experiments with normal hydrogen. Cooling down the trap slows down the reaction and leads, at a nominal trap temperature of 11 K, to a rate coefficient below 10⁻¹⁴ cm³ s⁻¹ for para-enriched hydrogen. The fitted reaction endothermicity of 25.3 ± 2.2 meV agrees well with the literature value calculated in the Born-Oppenheimer approximation, ΔH⁰ = 24.0 meV. A simpler evaluation procedure, fitting the data with Arrhenius functions, results in ^pk = 16.8 × 10⁻¹¹ exp(−234 K/T) cm³ s⁻¹ for pure para-hydrogen and ^ok = 9.4 × 10⁻¹¹ exp(−101 K/T) cm³ s⁻¹ for pure orthohydrogen.