= Solution
Distinguish the depth-dependent mantle <temperature> from <mantle potential temperature> $T_p$: the latter is the temperature a parcel would have after reversible decompression in an <adiabatic process> to a chosen reference pressure. A convecting mantle is approximately adiabatic through much of its interior, with steep <thermal boundary layers> near the surface and the <core-mantle boundary>. It is neither isothermal nor wholly molten.
Early <accretion>, large impacts and gravitational <planetary differentiation> supplied heat; metal segregation released gravitational energy and extensive melting could produce a <magma ocean>. Rapid surface heat loss then crystallized much of the silicate mantle. Subsequent evolution reflects competition between <radiogenic heating>, heat entering from the core and heat carried out through the upper boundary. In a schematic mantle <energy balance>,
$$
C_m\frac{d\overline T}{dt}=H_m(t)+Q_{\rm cmb}(t)-Q_m(t),\qquad
H_m(t)=\sum_jH_{j,0}e^{-\lambda_j t}.
$$
Here $C_m$ is mantle <heat capacity>, $H_m$ mantle radioactive power and $Q_m$ heat leaving the mantle; continental crustal radioactive production is not included in $H_m$. The convective <Urey ratio> $H_m/Q_m$ compares internal production with mantle heat loss. A value below one does not alone prove cooling, because core heat also enters; cooling requires $H_m+Q_{\rm cmb}<Q_m$. <Radioactive decay> reduces the heating supply, while <mantle convection>, <plate tectonics>, melt extraction and changing <viscosity> control the heat loss. A hotter mantle does not automatically imply proportionally faster plate motion: melting, dehydration and boundary strength can alter that relationship.
\Image[/past-exam-of-the-plate-tectonics-course-of-the-university-of-oxford/2022-mantle-thermal-history.png]
{title=Original schematic thermal histories and mantle geotherms; curves illustrate alternatives rather than reproduce observations or fit a particular thermal model}
The robust long-term picture is an energetic early <Earth> followed by substantial cooling and a presently mostly solid, convecting <mantle>. The shape of the solid-state history is less certain. Petrological reconstructions commonly place late-Archean ambient $T_p$ above modern values. One reconstruction infers about $1500$–$1600\,{}^{\circ}\mathrm{C}$ at $2.5$–$3$ Ga versus about $1350\,{}^{\circ}\mathrm{C}$ today, and permits warming before that late-Archean maximum after the initial magma-ocean cooling.<Thermal history of the Earth and its petrological expression by Claude Herzberg and others>[Abstract and discussion of the late-Archean temperature maximum, EPSL 292, pp. 79–88]{ref} This is a model-dependent ambient-mantle history, not proof that the accreting <Earth> was cooler than the Archean mantle or that every mantle reservoir followed one curve. \b[An early molten transient, subsequent solid-state heating or cooling, and later <secular cooling> must be distinguished.]
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