= Solution
The compositions alone do not specify a unique <thermodynamic activity> model. Adopt the usual ideal, independent <mixing on crystallographic sites> approximation: three Mg–Fe positions per formula unit, with the other sites fixed at their stated endmember compositions. Then the <mineral endmember> activities are
$$
a_{\rm phlogopite}=0.85^3,\qquad a_{\rm pyrope}=0.23^3.
$$
The exponent counts the three mixing cations; treating each whole mineral formula as a single mixing site would be a different model. In particular, the magnesian <biotite> and <garnet> components here are <phlogopite> and <pyrope>.
For the written reaction, the <reaction quotient> is
$$
Q=\frac{a_{\rm pyrope}}{a_{\rm phlogopite}}\frac{P}{P_0},
$$
with the other solid activities equal to one. The <Gibbs energy of reaction> balance therefore gives
$$
\boxed{T=\frac{130862.28}
{170.7-8.3145\ln\!\left[4500(0.23/0.85)^3\right]}
=981.235\,\mathrm K\simeq708.085\,{}^\circ\mathrm C.}
$$
The smaller product-endmember activity lowers the reaction <chemical potential> and lowers the equilibrium temperature in this model. Nonideal <activity coefficients> or unequal occupancies of crystallographic sites would require further information rather than a revised universal numerical answer.<Geochemistry by William M. White>[Online teaching draft, chapter 3, Example 3.4: pyrope activity as a product of site fractions, $a_{\rm pyrope}=X_{\rm Mg}^3X_{\rm Al}^2$. Here the aluminium sites are fixed at their endmember occupancy.]{ref}
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