The thought occurred to me that if we just happened to be born into a time when the the universe started slowing down, it would take some time for us to observe this - especially in more distant galaxies. Locally, we might observe attractive or static systems of galaxies. But because the light takes so long to reach us, more distant galaxies would still appear to be speeding away or even accelerating while they could have actually slowed, stopped, or even reversed velocity! Is this possible? Are there other data to confirm the accelerating expansion of the universe that are fresher/closer?
If we want to observe the universe expansion with the most straightforward and direct way we have to measure redshifts of galaxies. The minimum distance where this effect would start to be observable is that at which the speed of recession is larger than the average noise speed, which is around a few hundred $km/s$. Given that the value of the hubble constant is around $70 (km/s)/Mpc$ galaxies should be at least of the order of $10Mpc$ away for speed of recession not to be overshadowed by the noise, which is around $30$ million light years, so that is how recent the observation would be, $30$ million years old (which is very up to date in cosmological terms). Notice that if the hubble constant was slower, then we'd had to look further away and further back in time.
However if we want to trust general relativity (no reason not to) and use FRW equation and consider physically possible cosmological evolutions, we can use standard candles to measure acceleration/deceleration of the universe. That is to observe the apparent luminosity and compare it to the expected luminosity to deduce what distance has the light traveled since the supernova explosion until now, and compare that with its redshift predicted by FRW equations, this gives a measure of the cosmological constant, and assuming it is indeed a constant, the evolution of the universe is fixed by knowing its matter/dark energy etc constituents using general relativity. So that we know then that it must be NOW expanding or decelerating etc... But if you want to allow for crazy possibilities of GR breakdown, or violation of other basic observations about the universe which allows the universe to change its evolution even thought we know its constituents, then the naive empirical estimate given above should be a limiting factor.
This article suggests that an area of investigation might be the possibility that accelerating expansion of the universe is an illusion created by constant expansion. The article explains how if you are in a spaceship initially moving away from a start and then accellerate at a constant rate (eg 1g) toward the star, the star will look like it accellerates away from you, before decellerating and coming to a stop, then start getting closer to you (see the parabolic-looking graph at the bottom). Its suggested that these kinds of non-intuitive relativistic effects might explain the accelerating expansion of the universe.