The power is a continuous function of the parameter. If a statistical test has a power of 0.8, does that mean the power function is 0.8 at all parameters except the null hypothesis? Or does it mean the power function is almost everywhere 0.8 or at most 0.8?
You'll hear "this test has 80% power" as shorthand for a better statement like: "under a bunch of assumptions, including but not limited to this particular sample size and this particular true effect size, this test has an 80% probability of rejecting the null hypothesis with a two-sided alternative at a 5% significance level". Don't try to make sense of a statement like "this test has 80% power" unless there is a lot more detail provided. Any statement about power that is four words long is leaving out so much detail that it's effectively meaningless.
Statistical power is a function of statistical test, acceptable type I error rate, sample size, and effect size. It also depends on adherence to assumptions made about the data, such as assumptions of normality, independence of observations, etc, but often in a power analysis these are taken to be as presumed by the test used.
It seems like you are asking if the power is the same for all possible differences: no, the effect size has a strong influence on power for a given sample size.