555 IC Astable Mode - Does the voltage generally affect the frequency?
Voltage is not part of the equation, and that is because the voltage does not matter.

I confirm that the default voltage references are 1/3 and 2/3 of the VCC.
The RC time is also constant.  In electronics we can use RC time simply because it does not depend on voltage when using a constant voltage accros the R+C components in series.

When the voltage scales, all other properties scale accordingly, including the compare levels.

When VCC increases from 5V to 9V, the capacitor will charge faster from 1.667V to 3.333V, but it will require the same time to charge from \$\frac{VCC}{3}\$ to \$\frac{{2}{VCC}}{3}\$.

The RC time can be used as follows: if your capacitor is charged at V0, and you apply Vtot accross the RC series circuit, then your final voltage will be V0+0.63*(Vtot-V0) after a delay of RC.  The 63% percentage is constant.  You can replace Vtot with VCC and V0 with VCC/3.  After one RC, the voltage change is \$\frac{VCC}{3}+\frac{{0.63}{2}}{3}{VCC}= (\frac{1}{3}+\frac{{0.63}{2}}{3}){VCC}\$ = about 75% of VCC.  So, expressed as a % of VCC, the voltage change is constant and it will always reach 2/3 of VCC in the same delay.