Actually, you're right: the Hubble constant is not really constant. At least, it's not constant in time.
The reason it's called a constant is that, when Edwin Hubble originally compared the recession velocities of galaxies with their distances in 1929, there was no reason to expect any particular pattern. After all, just a few years prior, people had thought there were no other galaxies. But what Hubble found was that, except for a small amount of random variation, the velocities of galaxies were proportional to their distances; in other words, the ratio $v/d$ was roughly the same for all galaxies he observed. The value of this ratio came to be known as Hubble's constant, $H_0 \equiv \frac{v}{d}$, because it was constant from one galaxy to the next, rather than varying randomly as one might have guessed at the time.
Of course, it wasn't long before people realized that if the recessional velocity of each galaxy was proportional to its distance, you could extrapolate back to some point in the past at which $d = 0$: all the galaxies would have started out in the same place. This gives you an effective age for the universe. If you use a simple linear extrapolation, from basic kinematics you get
$$t = \frac{d}{v} = \frac{1}{H_0}$$
So the Hubble "constant" is not constant in time, but rather is inversely related to the age of the universe. As the universe gets older, the Hubble constant gets smaller, as you would expect. This happens because the distance $d$ to any given galaxy increases with time.
However, the fact that the universe has an age doesn't create an absolute time. Sure, different observers at different points in spacetime will measure different values for the age of the universe. And sure, you could define a time coordinate system by specifying that the time coordinate for any observer is the age of the universe as measured by that observer. This is called the comoving time, and it is a useful and sensible way to set up a coordinate system in time. But it is not the only possibility, and there is certainly nothing so special about it that it deserves to be called "absolute." Any observer who is moving with respect to the universe as a whole (i.e. relative to the Hubble flow) would not measure time at the same rate as this comoving time.
