We describe the state of development of a white light interferometer to characterize the cryogenic mirrors for GW detector on operation. We include the first experimental results from the proof of concept of the metrology instrument. The instrument will characterize the topology as well as the vibration of the mirrors. This development takes place in the frame of the E-TEST project. E-TEST is one of the technology demonstrators for the future Einstein Telescope (ET). ET is dedicated to the measure and characterization of gravitational waves. The prototype built by E-TEST includes a large silicon mirror of 40 cm diameter suspended by innovative vibration isolation hanging modules. To reach the detection specification, the mirror is cooled down at cryogenic temperatures around 20 K. Nevertheless, even after the isolation, the mirror may not reach perfect stability once at cryogenic temperatures. Furthermore, the mirror may experience surface topology changes and wavefront deformation due to the extreme variations in temperature and gradient. With our metrology instrument, we can obtain on a single camera frame a set of interferogram maps of the area observed on the mirror at different optical path differences. To do this, we design an innovative phase mask for a white light low-coherence interferometer. In addition, we implement new algorithms for the white light interferogram analysis, avoiding the limitations of the conventional Phase Shifting Interferometry algorithms.