How can I change the thickness of tick marks in BarLegend?

Somehow the AbsolutThickness you specified gets replaced by a default value of AbsoluteThickness[0.2].

This misbehavior can be corrected by replacing the incorrect value with your specification.

PlotLegends; (*preload definitions*)

Cell[BoxData[
   MakeBoxes@
      BarLegend[{"SunsetColors", {0, 1}}, LabelStyle -> {FontSize -> 12}, 
       Method -> {Frame -> False, TicksStyle -> Directive[Red, AbsoluteThickness[2]]}] /. 
     Directive[RGBColor[1, 0, 0], AbsoluteThickness[_]] -> 
      Directive[RGBColor[1, 0, 0], AbsoluteThickness[2]] // #[[1, 1]] &
   ], "Output"] // CellPrint

For opaque ticks:

Cell[BoxData[
   MakeBoxes@
      BarLegend[{"SunsetColors", {0, 1}}, LabelStyle -> {FontSize -> 12}, 
       Method -> {Frame -> False, TicksStyle -> Directive[Red, AbsoluteThickness[2]]}] /. 
     Directive[RGBColor[1, 0, 0], AbsoluteThickness[_]] -> 
      Directive[RGBColor[1, 0, 0], AbsoluteThickness[2], Opacity[1]] // #[[1, 1]] &
   ], "Output"] // CellPrint

---

Correcting the BarLegend of a DensityPlot, using the syntax provided in the answer by Praan :

DensityPlot[Sin[x y], {x, 0, 1}, {y, 0, 1}, 
  PlotLegends -> 
   BarLegend[Automatic, LabelStyle -> {FontSize -> 12}, 
    Method -> {Frame -> False, TicksStyle -> Directive[Red, AbsoluteThickness[2]]}]] /. 
 Placed[barLegend_BarLegend, args__] :> 
  Placed[ToExpression[
    FrameBox @@ MakeBoxes[barLegend] /. 
      Directive[Red, AbsoluteThickness[_]] -> 
       Directive[Red, AbsoluteThickness[2], Opacity[1]]], args]

The same output can be achieved by using the following LegendFunction

DensityPlot[Sin[x y], {x, 0, 1}, {y, 0, 1}, 
 PlotLegends -> 
  BarLegend[Automatic, LabelStyle -> {FontSize -> 12}, 
   Method -> {Frame -> False, 
     TicksStyle -> Directive[Red, AbsoluteThickness[2]]}, 
   LegendFunction -> (# /. 
       Directive[Red, AbsoluteThickness[_]] -> 
        Directive[Red, AbsoluteThickness[2], Opacity[1]] &)]]

---

With the answer by Praan and our discussion in the comments it became clear, that a wrong InterpretationFunction inside the TemplateBox created by BarLegend can cause additional problems.
Compare

MakeBoxes[
    BarLegend[{"SunsetColors", {0, 1}}, LegendMarkerSize -> 300, 
     LabelStyle -> {FontSize -> 12}, 
     Method -> {FrameStyle -> Black, AxesStyle -> None, 
       TicksStyle -> Black}]] /. 
   AbsoluteThickness[_] -> 
    AbsoluteThickness[2] /. (InterpretationFunction :> 
     f_) -> (InterpretationFunction :> (# &)) // ToExpression

with

MakeBoxes[
   BarLegend[{"SunsetColors", {0, 1}}, LegendMarkerSize -> 300, 
    LabelStyle -> {FontSize -> 12}, 
    Method -> {FrameStyle -> Black, AxesStyle -> None, 
      TicksStyle -> Black}]] /. 
  AbsoluteThickness[_] -> AbsoluteThickness[2] // ToExpression

or just the InterpretationFunction

MakeBoxes[
    BarLegend[{"SunsetColors", {0, 1}}, LegendMarkerSize -> 300, 
     LabelStyle -> {FontSize -> 12}, 
     Method -> {FrameStyle -> Black, AxesStyle -> None, 
       TicksStyle -> Directive[Black, AbsoluteThickness[2]]}]] /. 
   AbsoluteThickness[_] -> 
    AbsoluteThickness[2] // #[[-1, 2, 1]] & // ToExpression

and the first code block in the answer by Praan.

Update: You can also use the forms given in Options[Charting`iBarLegend] listed below, i.e., we can use Charting`TicksStyle for "TicksStyle", Charting`Frame for "Frame", etc.

BarLegend[{"SunsetColors", {0, 1}}, LabelStyle -> {FontSize -> 12}, 
 Charting`TicksStyle -> Directive[Opacity[1], Red, FontColor -> Black, AbsoluteThickness[5]],
 Charting`Frame -> False,
 Charting`TickLengths -> {20, 10}]

Note: This works for versions 9 thru 11 (Windows 10).

Original answer:

Use the (undocumented, afaik) options "TicksStyle" and "FrameStyle":

BarLegend[{"SunsetColors", {0, 1}}, LabelStyle -> {FontSize -> 12}, 
 "TicksStyle" -> Directive[Red, AbsoluteThickness[5]], 
 "FrameStyle" -> Opacity[0]]

In version 10, we need to add Opacity[1] in Directive[...]:

BarLegend[{"SunsetColors", {0, 1}}, LabelStyle -> {FontSize -> 12}, 
 "TicksStyle" -> Directive[Opacity[1], Red, FontColor->Black, AbsoluteThickness[5]], 
 "Frame" -> False]

A few more such options:

BarLegend[{"SunsetColors", {0, 1}}, 
 "LabelStyle" -> Directive[Green, Bold, FontSize -> Scaled[.4]],
  "Ticks" -> Range[0, 1, .1],
 "TicksStyle" -> Directive[Red, AbsoluteThickness[5]], 
 "FrameStyle" -> Directive[Dashed, Green], "TickSide" -> Left, 
 "LabelSide" -> Right, "Frame" -> True, "ImageSize" -> {100, 300}, 
 LegendMarkerSize -> {100, 500}]

Update: Full list of options

Options[Charting`iBarLegend]

Version 9:

Version 10:

Analysis

(Observations in 10.1.0 under Windows.)

Curiously it seems that AbsoluteThickness[0.2`] is hard-coded within the internal definitions. Formatting of BarLegend calls Charting`iBarLegend which calls Legending`LegendDump`iColorGradientLegend or Legending`LegendDump`iColorBandLegend. In the definition of iColorGradientLegend we find:

ticksstyle = Switch[ticksstyle,
  Automatic, {Directive[AbsoluteThickness[0.2`], GrayLevel[0.4`]], 
   Directive[AbsoluteThickness[0.2`], GrayLevel[0.4`], Opacity[0.3`]]},
  _?Charting`generalDirective | {_?
      Charting`generalDirective ..} | {}, (Charting`ConstructDirective[#1, 
      AbsoluteThickness[0.2`]] &) /@ Charting`padList[ticksstyle, 2],
  None | False, None,
  _, (Directive[AbsoluteThickness[0.2`], #1] &) /@ {GrayLevel[0.4`], 
    Directive[GrayLevel[0.4`], Opacity[0.5`]]}]

If the option value passed to this inner function is a "generalDirective" we specifically get:

(Charting`ConstructDirective[#1, 
  AbsoluteThickness[0.2`]] &) /@ Charting`padList[ticksstyle, 2]

So the program is appending AbsoluteThickness[0.2`] to anything we specify by way of Charting`ConstructDirective. This seems at best a poor design in my opinion.

Work-around

Seeing the cause of the problem inspires an unusual work-around; here is a chomp head that consumes the argument to its right and then vanishes:

chomp /: h_[a___, chomp[x__], b_, c___] := h[a, x, c]

Now when used on our AbsoluteThickness spec making sure it is the last:

BarLegend[{"SunsetColors", {0, 1}}, LabelStyle -> {FontSize -> 12}, 
 Method -> {Frame -> False, 
   TicksStyle -> Directive[Red, Opacity[1], chomp @ AbsoluteThickness[4]]}]

One could also enter Directive[chomp[ Red, Opacity[1], AbsoluteThickness[4] ] ].