This happens because PGFPlots only uses one "stack" per axis: You're stacking the second confidence interval on top of the first. The easiest way to fix this is probably to use the approach described in "Is there an easy way of using line thickness as error indicator in a plot?": After plotting the first confidence interval, stack the upper bound on top again, using stack dir=minus
. That way, the stack will be reset to zero, and you can draw the second confidence interval in the same fashion as the first:

\documentclass{standalone}
\usepackage{pgfplots, tikz}
\usepackage{pgfplotstable}
\pgfplotstableread{
temps y_h y_h__inf y_h__sup y_f y_f__inf y_f__sup
1 0.237340 0.135170 0.339511 0.237653 0.135482 0.339823
2 0.561320 0.422007 0.700633 0.165871 0.026558 0.305184
3 0.694760 0.534205 0.855314 0.074856 -0.085698 0.235411
4 0.728306 0.560179 0.896432 0.003361 -0.164765 0.171487
5 0.711710 0.544944 0.878477 -0.044582 -0.211349 0.122184
6 0.671241 0.511191 0.831291 -0.073347 -0.233397 0.086703
7 0.621177 0.471219 0.771135 -0.088418 -0.238376 0.061540
8 0.569354 0.431826 0.706882 -0.094382 -0.231910 0.043146
9 0.519973 0.396571 0.643376 -0.094619 -0.218022 0.028783
10 0.475121 0.366990 0.583251 -0.091467 -0.199598 0.016664
}{\table}
\begin{document}
\begin{tikzpicture}
\begin{axis}
% y_h confidence interval
\addplot [stack plots=y, fill=none, draw=none, forget plot] table [x=temps, y=y_h__inf] {\table} \closedcycle;
\addplot [stack plots=y, fill=gray!50, opacity=0.4, draw opacity=0, area legend] table [x=temps, y expr=\thisrow{y_h__sup}-\thisrow{y_h__inf}] {\table} \closedcycle;
% subtract the upper bound so our stack is back at zero
\addplot [stack plots=y, stack dir=minus, forget plot, draw=none] table [x=temps, y=y_h__sup] {\table};
% y_f confidence interval
\addplot [stack plots=y, fill=none, draw=none, forget plot] table [x=temps, y=y_f__inf] {\table} \closedcycle;
\addplot [stack plots=y, fill=gray!50, opacity=0.4, draw opacity=0, area legend] table [x=temps, y expr=\thisrow{y_f__sup}-\thisrow{y_f__inf}] {\table} \closedcycle;
% the line plots (y_h and y_f)
\addplot [stack plots=false, very thick,smooth,blue] table [x=temps, y=y_h] {\table};
\addplot [stack plots=false, very thick,smooth,blue] table [x=temps, y=y_f] {\table};
\end{axis}
\end{tikzpicture}
\end{document}
The label with " .10^-3" is called "tick scale label" and the associated style can be modified using colorbar style={y tick scale label style={...}}
:
\documentclass{standalone}
\usepackage{pgfplots}
\pgfplotsset{compat=1.10}
\begin{document}
\begin{tikzpicture}
\begin{axis}[colorbar,colorbar style={y tick scale label style={xshift=0.5cm}}]
\addplot[mesh,ultra thick] {20000*x};
\end{axis}
\end{tikzpicture}
\end{document}

EDIT
In order to align the tick scale label in the same way as tick labels, you can use
y tick scale label style={
at={(yticklabel* cs:1,\pgfkeysvalueof{/pgfplots/major tick length})},
yshift=7pt,
anchor=near yticklabel,
},
instead of the absolute shift value. This will place it exactly as the tick labels. The point (yticklabel* cs:1)
is always the uppermost point on the axis, and :\pgfkeysvalueof{/pgfplots/major tick length}
shifts it by the default amount just as tick labels.
Best Answer
Any colorbar style change will be redefined by colorbar horizontal and the alikes. Thus the new style must be placed AFTER colorbar horizonal.
For more information, please refer to its manual p 140-142.