Einstein's elevator thought experiment contains the following premise:
An observer in a box will see free bodies moving identically between when the box is uniformly accelerating in gravitational free space and when the box is at rest in homogeneous gravitational field. How can one empirically check this premise to verify or falsify it? Is this based on special relativity?
This thought experiment is an example of the equivalence principle, which is the key part of general relativity that differentiates it from other relativistic theories of gravity, such as Brans-Dicke gravity. Many methods have been found for testing the equivalence principle. A good review article on this kind of thing is Will, "The Confrontation between General Relativity and Experiment," https://arxiv.org/abs/1403.7377 . There are both laboratory experiments, such as Eotvos experiments, and solar system tests.
Is this based on special relativity?
No, the equivalence principle specifically talks about gravity, and gravity isn't described by special relativity. However, the e.p. does imply that certain experiments in gravitational fields can be described correctly by SR. A good example is the Pound-Rebka experiment.
