In the brief time since the outbreak of the COVID-19 pandemic, and despite its proofreading mechanism, the SARS-CoV-2 coronavirus has accumulated a significant amount of genetic variability through recombination and mutation events. To test evolutionary trends that could inform us on the adaptive process of the virus to its human host, we summarize all this variability in the Sequence Compositional Complexity (SCC), a measure of genome heterogeneity that captures the mutational and recombinational changes accumulated by a nucleotide sequence along time. Despite the brief time elapsed, we detected many differences in the number and length of compositional domains, as well as in their nucleotide frequencies, in more than 12,000 high-quality coronavirus genomes from across the globe. These differences in SCC are phylogenetically structured, as revealed by significant phylogenetic signal. Phylogenetic ridge regression shows that SCC followed a generalized decreasing trend along the ongoing process of pathogen evolution. In contrast, SCC evolutionary rate increased with time, showing that it accelerates toward the present. In addition, a low rate set of genomes was detected in all the genome groups, suggesting the existence of a stepwise distribution of rates, a strong indication of selection in favor of different dominant strains. Coronavirus variants reveal an exacerbation of this trend: non-significant SCC regression, low phylogenetic signal and, concomitantly, a threefold increase in the evolutionary rate. Altogether, these results show an accelerated decline of genome heterogeneity along with the SARS-CoV-2 pandemic expansion, a process that might be related to viral adaptation to the human host, perhaps paralleling the transformation of the current pandemic to epidemic.