Surface polishing by laser remelting (SP-LRM aka laser polishing) is a material finishing process that uses the thermal energy of a laser beam tracked across a surface remelting a thin surface layer at the laser incidence. This causes formation of thermodynamic molten material flows within a melt pool leading to the redistribution of the initial surface topography and bulk material with hills flowing into valleys. The effectiveness of this process is highly dependent on a complex combination of the laser-motion-material parameters (e.g., power, speed, focus offset, etc.) and their formation of thermodynamic flows in the laser-material interaction zone. In an ideal case, the chosen process parameter combination forms a stable thermodynamic equilibrium between applied laser energy and convection, conduction and radiation being the triplet of thermal energy components. It is assumed that this stable thermodynamics of laser-material interaction leads to the formation of uniform (in terms of material redistribution) laser remelted tracks. Thermodynamic instabilities on the contrary may cause sudden variations and non-uniformities in the surface topography, such as bulges, cavities, and wide/narrow stripes, negatively affecting the smoothening effect of surface polishing. This study demonstrates the formation of uniform laser tracks from a stable LRM process and non-uniform laser tracks from a non-stable LRM process through the use of statistical analysis of the surface topography measured off-line by white-light interferometry and in-situ thermographic imaging. Through a combination of these techniques, multiple examples of regions containing different types of these undesired surface topography non-uniformities are identified. One of the goals of this study was to find an interdependence between laser-processed surface topography and thermographic response. For this purpose, information data have been synchronized in time and space domain. In detail, stable and non-stable LRM process states, typical thermodynamic instabilities, initial surface defects, and especially the beginning and ending of LRM tracks are analyzed. The results from this preliminary study lay a scientific and engineering basis to better understand the thermodynamics of the LRM process and identify statistical metrics capable of recognizing thermodynamic imbalance for future development of online LRM process monitoring, control, and optimization.