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The title says it all. Why is the Hubble constant written $H_0$ and not just $H$? What is the purpose of that subscript $_0$?

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The Hubble parameter is defined as $\frac{\dot{a}}{a}$, where $a$ is the scale factor in the Friedmann-Robertson-Walker metric: $$g_{\mu\nu} \, \mathrm{d}x^{\mu} \, \mathrm{d}x^{\nu}=c^2 \, \mathrm{d}t^2-a^2 \left(\frac{\mathrm{d}r^2}{1-kr^2}+r^2\mathrm{d}\Omega ^2 \right) \,.$$

When you put this metric into Einstein Field Equation, you will get an equation that describes how $a$ evolves with time $t$. Thus in general the Hubble parameter is a function of time.

$H_0$ is used to indicate the actual value of the Hubble parameter, measured at our time.

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The Hubble "constant" is not a constant; it changes with time and should be referred to as the Hubble parameter. You can find a discussion/plots of this variation with time at How does the Hubble parameter change with the age of the universe?

The value of the Hubble parameter at the present epoch is represented with $H_0$.

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