  For the $p$-cyclotomic tower of $\mathbb{Q}_p$ Fontaine established a
description of local Iwasawa cohomology with coefficients in a local Galois
representation $V$ in terms of the $\psi$-operator acting on the attached etale
$(\varphi,\Gamma)$-module $D(V)$. In this article we generalize Fontaine's
result to the case of arbitratry Lubin-Tate towers $L_\infty$ over finite
extensions $L$ of $\mathbb{Q}_p$ by using the Kisin-Ren/Fontaine equivalence of
categories between Galois representations and $(\varphi_L,\Gamma_L)$-module and
extending parts of [Herr L.: Sur la cohomologie galoisienne des corps
$p$-adiques. Bull. Soc. Math. France 126, 563-600 (1998)], [Scholl A. J.:
Higher fields of norms and $(\phi,\Gamma)$-modules. Documenta Math.\ 2006,
Extra Vol., 685-709]. Moreover, we prove a kind of explicit reciprocity law
which calculates the Kummer map over $L_\infty$ for the multiplicative group
twisted with the dual of the Tate module $T$ of the Lubin-Tate formal group in
terms of Coleman power series and the attached $(\varphi_L,\Gamma_L)$-module.
The proof is based on a generalized Schmid-Witt residue formula. Finally, we
extend the explicit reciprocity law of Bloch and Kato [Bloch S., Kato K.:
$L$-functions and Tamagawa numbers of motives. The Grothendieck Festschrift,
Vol. I, 333-400, Progress Math., 86, Birkh\"auser Boston 1990] Thm. 2.1 to our
situation expressing the Bloch-Kato exponential map for $L(\chi_{LT}^r)$ in
terms of generalized Coates-Wiles homomorphisms, where the Lubin-Tate
characater $\chi_{LT}$ describes the Galois action on $T.$
