The first thing I would recommend would be to see whether any covariates potentially related to outcome changed over the time course from 2000 to 2015. For example, was some new type of therapy introduced over that time, or did measures of performance status of newly enrolled patients change over time? That will given an idea whether the differences in mortality are with respect to time of study entry per se or represent a change over time in characteristics of the patients enrolling in the study.
Then you can include the year (or time period) of entry into the study as a covariate in a Cox proportional hazards regression, along with standard clinical covariates. That will help control for covariates that might have changed in the enrolling population from 2000 to 2015. It might make sense to model year of entry as a continuous predictor, modeled flexibly with restricted cubic splines, rather than breaking year of entry up into arbitrary groups. See this page among others on this site for why binning a continuous predictor can be problematic.
Provided that the proportional hazards assumption is adequately respected, a significant coefficient for year (or time period) of entry would indicate a significant relationship between year of study entry and mortality. Furthermore, such a result with a Cox model would document that relationship while controlling for the other covariates.