I suspect there is a typo in the linearization of the Tv^4 into Tv^{n+1} and Tv^{n} in equation 2.4.14 The typo is because n in equation 2.4.14 refers to time step. In contrast, n in the text following equation 5-129 refers to iteration level. I believe the linearization of the Tv^4 in equation 2.4.14 should be instead written as

$\sigma (T_v^{n+1,k})^3 [ T_v^{n+1,k} + 4 (T_v^{n+1,k+1} - T_v^{n+1})]$
$= \sigma (T_v^{n+1,k})^3 [ T_v^{n+1,k} + 4 (\Delta T_v^{n+1,k})]$

where $k$ is the iteration level. If you do make such a change to equation 2.4.14, you should consider using $k$ instead of $n$ in Section 2.5.3.2 for consistency.

One may be inclined to change the text following equation 2.4.14 to say n and n+1 are iteration levels instead of time steps. But, such a change would be incorrect because the ground temperature Tg^{n} and Tg^{n+1} must correspond to previous and current time steps (as explained in detail in Section 7.3 of Grodon's Climate Change and Terrestrial Ecosystem Modeling textbook).