I have a binarized image and would like to set pixel values which are 1 to a certain (any) color picked from ColorData["TemperatureMap"] and 0 to another color of "TemperatureMap".
How can I do that?

Thanks in advance. Milenko
Update: Using EventHandler and TemperatureMap:
image = Import["https://i.sstatic.net/qmKND.png"]; DynamicModule[{col = Green}, EventHandler[Column[{ColorData["TemperatureMap", "Image"], Dynamic@Colorize[image, ColorRules -> {0 -> White, 1 -> col}, ImageSize -> 300]}], {"MouseClicked" :> (col = ColorData["TemperatureMap"][First[MousePosition["Graphics"]]])}, PassEventsDown -> Automatic, PassEventsUp -> True]] 
Update 2: Select background color using right-mouse-click:
DynamicModule[{col1 = Green, col2 = White}, EventHandler[Column[{ColorData["TemperatureMap", "Image"], Dynamic@Colorize[image, ColorRules -> {0 -> col2, 1 -> col1},ImageSize -> 300]}], {{"MouseClicked", 1} :> (col1 = ColorData["TemperatureMap"][First[MousePosition["Graphics"]]]), {"MouseClicked", 2} :> (col2 = ColorData["TemperatureMap"][First[MousePosition["Graphics"]]])}, PassEventsDown -> Automatic, PassEventsUp -> True]] 
Update 3:
I want to pick up a special color from the color table "without GUI" and set all intensities of 1 with this color ...
col1 = ColorData["SunsetColors"][.7]; col2 = ColorData["SunsetColors"][.1]; Colorize[image, ColorRules -> {0 -> col2, 1 -> col1}, ImageSize -> 300] 
Original answer:
image = Import["https://i.sstatic.net/qmKND.png"]; Column[{ColorSetter[Dynamic[x]], Dynamic@Colorize[MorphologicalComponents@image, ColorRules -> {0 -> White, _ :> x}]}] 
Clicking on the ColorSetter button invokes color selection dialog:

After selecting a color, the colors in the image are updated:

col1 = ColorData["TemperatureMap"][.2]; col2 = ColorData["TemperatureMap"][.6]; Colorize[image, ColorRules -> {0 -> col2, 1 -> col1}, ImageSize -> 300]? $\endgroup$