  Let $p$ be an odd prime number. In this paper, we study the growth of the
Sylow $p$-subgroups of the even $K$-groups of rings of integers in a $p$-adic
Lie extension. Our results generalize previous results of Coates and Ji-Qin,
where they considered the situation of a cyclotomic $\mathbb{Z}_p$-extension.
Our method of proof differs from these previous work. Their proof relies on an
explicit description of certain Galois group via Kummer theory afforded by the
context of a cyclotomic $\mathbb{Z}_p$-extension, whereas our approach is via
considering the Iwasawa cohomology groups with coefficients in
$\mathbb{Z}_p(i)$ for $i\geq 2$. We should mention that this latter approach is
possible thanks to the Quillen-Lichtenbaum Conjecture which is now known to be
valid by the works of Rost-Voevodsky. We also note that the approach allows us
to work with more general $p$-adic Lie extensions that do not necessarily
contain the cyclotomic $\mathbb{Z}_p$-extension, where the Kummer theoretical
approach does not apply. Along the way, we study the torsionness of the second
Iwasawa cohomology groups with coefficients in $\mathbb{Z}_p(i)$ for $i\geq 2$.
Finally, we give examples of $p$-adic Lie extensions, where the second Iwasawa
cohomology groups can have nontrivial $\mu$-invariants.
