= Solution
Adopt ideal independent <mixing on crystallographic sites>, with Mg and Fe sharing three positions per formula unit and the other endmember sites fixed. This gives the <phlogopite> and <pyrope> activities
$$
a_{\rm phlogopite}=0.80^3,\qquad a_{\rm pyrope}=0.20^3.
$$
The <reaction quotient> contains their ratio, since <pyrope> is a product and <phlogopite> a reactant. Consequently
$$
\boxed{T=\frac{130862.28}
{170.7-8.3145\ln[4500(0.20/0.80)^3]}
=966.923\,\mathrm K\simeq693.773\,{}^\circ\mathrm C.}
$$
The site-mixing assumption is part of this numerical answer: compositions alone do not determine nonideal <thermodynamic activities>. More detailed <activity coefficients> or unequal site occupancies would need an additional model. The cube counts the three magnesium-bearing positions rather than three molecules of each mineral.
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