  A recent experiment [J.L. Garrett et al., Phys. Rev. Lett {\bf 120}, 040401
(2018)] measured for the first time the gradient of the Casimir force between
two gold spheres at room temperature. The theoretical analysis of the data was
carried out using the standard Proximity Force Approximation (PFA). A fit of
the data, using a parametrization of the force valid for the sphere-plate
geometry, was used by the authors to place a bound on deviations from PFA.
Motivated by this work, we compute the Casimir force between two gold spheres
at finite temperature. The semi-analytic formula for the Casimir force that we
construct is valid for all separations, and can be easily used to interpret
future experiments in both the sphere-plate and sphere-sphere configurations.
We describe the correct parametrization of the corrections to PFA for two
spheres that should be used in data analysis.
