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The uniaxial pressure dependence of the critical temperature causes a reversible effect of strain on the critical current density and the flux pinning strength in many high-temperature superconductors. Recent experiments on patterned coated conductor bridges have shown that the anisotropic nature of the pressure dependence of the critical temperature of rare earth (RE)-Ba2Cu3O7-δ (REBCO) has a major impact on the performance of coated conductors under strain. The strain effect on the critical current density is most prominent when the strain is along the  and  directions of the superconducting film, whereas it almost completely disappears when the strain is along . In this paper, we investigate the correlation between the uniaxial-pressure dependence of the critical temperature and the reversible strain effect on flux pinning in REBCO coated conductors. We show that axial strain has a large effect on the irreversibility field and the pinning force in coated conductors when the  and  directions of the superconducting film are aligned along the conductor axis. The magnitude of the strain effect in these conductors largely depends on the angle at which the magnetic field is applied. On the other hand, the critical temperature is not expected to change with the axial strain in coated conductors when the  direction is aligned along the conductor axis. Indeed, the irreversibility field and the magnetic field dependence of the pinning force of these conductors are almost independent of the axial strain for all angles at which the magnetic field is applied. The minor strain dependence of the critical current measured in these conductors could be caused by the average in-plane grain misalignment of between 6° and 8°, which causes a slight variation in the strain alignment with the axes of the superconducting film. The results confirm that the reversible strain effect- in REBCO coated conductors is largely determined by the uniaxial pressure dependence of the critical temperature.