Hubble time, the age of the Universe and expansion rate

It is in fact a reflection of the fact that the rate of expansion has been nearly constant for a long time.

Mathematically, the expansion of the universe is described by a scale factor $a(t)$, which can be interpreted as the size of the universe at a time $t$, but relative to some reference size (typically chosen to be the current size). The Hubble parameter is defined as

$$H = \frac{\dot{a}}{a}$$

and the Hubble time is the reciprocal of the Hubble parameter,

$$t_H = \frac{a}{\dot{a}}$$

Now suppose the universe has been expanding at a constant rate for its entire history. That means $a(t) = ct$. If you calculate the Hubble time in this model, you get

$$t_H = \frac{ct}{c} = t$$

which means that in a linear expansion model, the Hubble time is nothing but the current age of the universe.

In reality, the best cosmological theories suggest that the universe has not been expanding linearly since the beginning. So we would expect that the age of the universe is not exactly equal to the Hubble time. But hopefully it makes sense that if any nonlinear expansion lasted for only a short period, then the Hubble time should still be close to the age of the universe. That is the situation we see today.

For more information on this, I'd suggest you check out these additional questions

• Value of the Hubble parameter over time
• Universe Expansion as an absolute time reference

and others like them.

It is a coincidence.

The reason is that the Hubble constant H is not constant, and varies over time. For example 6 billion years ago, when the universe was 7.5 billion years old, the Hubble constant was about 100 (km/s)/Mpc, what means the Hubble time was 9.78 billion years. When the universe is 24 billion years of age, H will be 60 (km/s)/Mpc, and the Hubble time will be 16.3 billion years.

Not even close to the age of the universe.