  We generalize $f(R)$-theory of anti-evaporation/evaporation for a
Reissner-Nordstr\"om black hole in $n$-dimensional space-time. We consider
non-linear conformally invariant Maxwell field. By perturbing the fields over
Nariai-like space-time associated with degenerate horizon, we describe
dynamical behavior of horizon. We show that $f(R)$-gravity can offer both
anti-evaporation and evaporation in $n$-dimensional Reissner-Nordstr\"om black
hole depending on the dimension $n$ and the functional form of $f(R)$.
Furthermore, we argue that, in one class of non-oscillatory solution, stable
and unstable anti-evaporation/evaporation exist. In the other class of
oscillatory solution anti-evaporation/evaporation exists only with instability.
The first class of solution may explain a long-lived black hole.
