Biological process

positive regulation of CD8-positive T lymphocyte differentiation, positive regulation of CD8-positive T-cell differentiation, positive regulation of CD8-positive T-lymphocyte differentiation, positive regulation of CD8-positive, alpha beta T lymphocyte differentiation, positive regulation of CD8-positive, alpha beta T-cell differentiation, positive regulation of CD8-positive, alpha beta T-lymphocyte differentiation, up regulation of CD8-positive, alpha-beta T cell differentiation, up-regulation of CD8-positive, alpha-beta T cell differentiation, upregulation of CD8-positive, alpha-beta T cell differentiation

Any process that activates or increases the frequency, rate or extent of CD8-positive, alpha-beta T cell differentiation.

positive regulation of CD8-positive, alpha-beta T cell development

Note that immunologists typically use the word 'development' to refer to cells of B or T cell lineages undergoing the process that GO describes as 'cell differentiation'.