A better solution is

from matplotlib.ticker import FuncFormatter

fmt = lambda x, pos: '{:.1%}'.format(x)
cbar = plt.colorbar(format=FuncFormatter(fmt))

Update the ticks and the tick labels:

cbar.set_ticks([mn,md,mx])
cbar.set_ticklabels([mn,md,mx])

As explained in the comments by Ernest, the colorbar for a contour gets a FixedFormatter, which doesn't accept useMathText. In my version, the code didn't give an error, but as you pointed out, also didn't generate the desired output.

So, the answer is to change the formatter to a ScalarFormatter, and at the same time tell it to use math text. The formatter for a vertical colorbar can be set via cbar.ax.yaxis.set_major_formatter.

from matplotlib import pyplot as plt
from matplotlib import ticker
import numpy as np

plt.tricontourf(np.random.uniform(1, 10, 100), np.random.uniform(1, 10, 100), np.random.uniform(1e-6, 1e-5, 100))
cbar = plt.colorbar()
cbar.ax.yaxis.set_major_formatter(ticker.ScalarFormatter(useMathText=True))
plt.show()

Please let me know whether it now works for you. (My system, now with matplotlib 3.2.1, Pycharm 2019.3.4 under Windows.)

ticklabel_format(..., scilimits=(m, n) can be used to force a scientific format for powers of 10 outside the range between m and n. With (0,0) scientific format will always be used.

If you are using a lognorm, the colorbar gets both major and minor ticks especially to show log formatting. You can change their format and their position to standard ticks first, as follows:

from matplotlib import pyplot as plt
from mpl_toolkits import axes_grid1
from matplotlib import ticker
from matplotlib import colors
import numpy as np

def add_colorbar(im, aspect=15, pad_fraction=0.5, **kwargs):
    """Add a vertical color bar to an image plot."""
    divider = axes_grid1.make_axes_locatable(im.axes)
    width = axes_grid1.axes_size.AxesY(im.axes, aspect=1./aspect)
    pad = axes_grid1.axes_size.Fraction(pad_fraction, width)
    current_ax = plt.gca()
    cax = divider.append_axes("right", size=width, pad=pad)
    plt.sca(current_ax)
    cbar = im.axes.figure.colorbar(im, cax=cax, **kwargs)
    cbar.ax.yaxis.set_major_locator(ticker.AutoLocator())
    cbar.ax.yaxis.set_minor_locator(ticker.AutoLocator())
    cbar.ax.yaxis.set_major_formatter(ticker.ScalarFormatter(useMathText=True, useOffset=True))
    cbar.ax.xaxis.set_major_formatter(ticker.ScalarFormatter())
    cbar.ax.ticklabel_format(style='sci', scilimits=(0, 0))
    return cbar

im = plt.imshow(np.random.uniform(8000, 12000, (10,10)), norm=colors.LogNorm())
cbar = add_colorbar(im)

plt.show()

You can keep the same locators as proposed by the colorbar function but change the ticklabels in order to print the formatted date as follows:

# change colobar ticks labels and locators 
cbar.set_ticks([s.colorbar.vmin + t*(s.colorbar.vmax-s.colorbar.vmin) for t in cbar.ax.get_yticks()])
cbar.set_ticklabels([mdates.datetime.datetime.fromtimestamp((s.colorbar.vmin + t*(s.colorbar.vmax-s.colorbar.vmin))/1000000000).strftime('%c') for t in cbar.ax.get_yticks()])
plt.show()

which gives the result below:

If you really want to control tick locations, you can compute the desired values (here for approximately 3 months intervals ~91.25 days):

i,ticks = 0,[s.colorbar.vmin]
while ticks[-1] < s.colorbar.vmax:
   ticks.append(s.colorbar.vmin+i*24*3600*91.25*1e9)
   i = i+1
ticks[-1] = s.colorbar.vmax
cbar.set_ticks(ticks)
cbar.set_ticklabels([mdates.datetime.datetime.fromtimestamp(t/1e9).strftime('%c') for t in ticks])

You are suffering from an off-by-one error. You have 10 ticklabels spread among 11 colors. You might be able to correct the error by using np.linspace instead of np.arange. Using np.linspace the third argument is the number of values desired. This reduces the amount of mental gymnastics needed to avoid the off-by-one error:

import matplotlib.pyplot as plt
import numpy as np
import matplotlib.cm as cm
import matplotlib.colors as mcolors

def colorbar_index(ncolors, cmap):
    cmap = cmap_discretize(cmap, ncolors)
    mappable = cm.ScalarMappable(cmap=cmap)
    mappable.set_array([])
    mappable.set_clim(-0.5, ncolors+0.5)
    colorbar = plt.colorbar(mappable)
    colorbar.set_ticks(np.linspace(0, ncolors, ncolors))
    colorbar.set_ticklabels(range(ncolors))

def cmap_discretize(cmap, N):
    """Return a discrete colormap from the continuous colormap cmap.

        cmap: colormap instance, eg. cm.jet. 
        N: number of colors.

    Example
        x = resize(arange(100), (5,100))
        djet = cmap_discretize(cm.jet, 5)
        imshow(x, cmap=djet)
    """

    if type(cmap) == str:
        cmap = plt.get_cmap(cmap)
    colors_i = np.concatenate((np.linspace(0, 1., N), (0.,0.,0.,0.)))
    colors_rgba = cmap(colors_i)
    indices = np.linspace(0, 1., N+1)
    cdict = {}
    for ki,key in enumerate(('red','green','blue')):
        cdict[key] = [ (indices[i], colors_rgba[i-1,ki], colors_rgba[i,ki])
                       for i in xrange(N+1) ]
    # Return colormap object.
    return mcolors.LinearSegmentedColormap(cmap.name + "_%d"%N, cdict, 1024)

fig, ax = plt.subplots()
A = np.random.random((10,10))*10
cmap = plt.get_cmap('YlGnBu')
ax.imshow(A, interpolation='nearest', cmap=cmap)
colorbar_index(ncolors=11, cmap=cmap)    
plt.show()