To know if the universe is expanding, we need to know the following: how far away things are, and whether they're moving away from us. In an accelerating universe, there's evidence that things were moving away from us slower in the past.
The evidence that things are moving away from us is that as the universe expands, the wavelength of any light travelling through it also expands, shifting it towards the red end of the spectrum: redshift. (Think of the wavelength of a wavy line drawn on a piece of rubber that's then stretched out.) We've known for nearly 100 years that the universe is expanding. (Edit: not just that the objects in it are moving apart but that space itself is expanding.)
The evidence of accelerating expansion is that based on the redshift of nearby objects, we would expect distant objects to have a higher redshift than they actually do. That means they were moving away from us slower in the past, so something must be accelerating them.
We use the brightness of Type Ia supernovae to measure distance. Even though supernovae aren't identically bright, their brightness follows a curve which lets us calculate their peak brightness (they are standardizable).
This paper argues that the calculations cosmologists use to standardize supernovae brightness fail to take into account the age of the progenitor stars, as far as I can tell. If true, this means our distance measurements are inaccurate and these stars are actually closer than we thought, enough to restore to linear relationship between distance and redshift that one would expect in a universe expanding at a constant rate.
In other words, their redshift is lower not because the expansion rate of space was lower in the past, but because they're not as far away as we thought they were.