Yes, the frequency of oscillation is independent of power supply voltage \$V_{CC}\$. Almost. It's impossible to make it completely independent, but the designers went to great lengths to keep \$V_{CC}\$ out of that equation.
This is because the 555 timer IC design relies on the time it takes for a capacitor to charge and discharge to some fraction of the supply, not to any particular specified voltage. Whether your supply voltage \$V_{CC}\$ is 6V or 12V, the timing capacitor will charge to \$\frac{2}{3}\$ of that supply in the same amount of time; it's just that by doubling the voltage, you also double the charging rate. The time constant \$\tau = R \times C\$ of the resistor+capacitor system remains the same in all cases:

simulate this circuit – Schematic created using CircuitLab
On the left the supply is +6V, and on the right it's double that at +12V, but the resistors R1 and R2 have the same values, as do capacitors C1 and C2. Therefore the time constant in both cases is the same:
$$ \tau = R \times C = 1k\Omega \times 1\mu F = 1ms $$
Here's a graph of the voltage across the capacitors, at X and Y as they charge, from the moment power is applied at time \$t=0s\$:

The horizontal green markers represent \$\frac{2}{3}\$ of the supplies, 4V and 8V. Notice how both outputs X and Y reach their respective \$\frac{2}{3}\$ point at exactly the same instant, \$t=1.1ms\$. Therefore, the time constant represents just that; the time it would take for the capacitor to charge to about \$\frac{2}{3}=67\%\$ of its final value. More precisely, it's 63.2%, but in any case, it's the same regardless of supply voltage.
There will be a tiny variation of oscillator frequency with \$V_{CC}\$, because there are other elements inside the IC which are not quite so independent of that supply, but that variation won't be nearly as significant as frequency variation with time and temperature, both of which cause capacitor and resistor values to change. This should be of greater concern.
If you want an oscillator that is truly immune to power supply voltage and component degradation, you should to use a crystal, which can guarantee oscillation at a frequency to within about 0.01% of of its rated frequency, over a larger temperature range, and a much longer lifespan.