Why does \\ not return a new line in an equation?

Both equation and equation* do not allow multi lined expressions. Use aligned from amsmath, inside equation* to break in to many lines. If alignment is not needed, you can use gathered (instead of aligned) as suggested by Mico.

\documentclass{article}
\usepackage{amsmath}
\begin{document}
  \begin{equation*}\label{eq:pareto mle2}
  \begin{aligned}
A_0 = \frac{1}{(\alpha+t_x)^{r+s+x}}{}_2F_1\left(r+s+x,x+1;r+s+x+1;\frac{\alpha-\beta}{\alpha + t_x} \right ) \\
 - \frac{1}{(\alpha+T)^{r+s+x}}{}_2F_1\left(r+s+x,x+1;r+s+x+1;\frac{\alpha-\beta}{\alpha + T} \right ),
 \end{aligned}
\end{equation*}
\end{document}

Or use multlined from mathtools:

\documentclass{article}
\usepackage{mathtools}
\begin{document}
  \begin{equation*}\label{eq:pareto mle2}
  \begin{multlined}
A_0 = \frac{1}{(\alpha+t_x)^{r+s+x}}{}_2F_1\left(r+s+x,x+1;r+s+x+1;\frac{\alpha-\beta}{\alpha + t_x} \right ) \\
 - \frac{1}{(\alpha+T)^{r+s+x}}{}_2F_1\left(r+s+x,x+1;r+s+x+1;\frac{\alpha-\beta}{\alpha + T} \right ),
 \end{multlined}
\end{equation*}
\end{document}

The amsmath package -- which you appear to be loading already as your example code uses an environment called equation* -- provides quite a few environments to typeset equations that need to be split across two or more lines. The align* environment may be particularly suitable to the case at hand; note the use of & to state the alignment points.

\documentclass{article}
\usepackage{amsmath}
\begin{document}
\begin{align*}\label{eq:pareto mle2}
A_0 &=   \frac{1}{(\alpha+t_x)^{r+s+x}}{}_2 F_1\left( r+s+x,x+1;r+s+x+1;\frac{\alpha-\beta}{\alpha + t_x} \right)\\
&\quad - \frac{1}{(\alpha+T)^{r+s+x}}{}_2 F_1\left( r+s+x,x+1;r+s+x+1;\frac{\alpha-\beta}{\alpha + T} \right),
\end{align*}
\end{document}

Addendum: Note the close similarity of the outputs produced by the single align* environment in this answer and by the nested pair of equation*/multlined environments in @HarishKumar's answer.

There is an easy way to make \\ works in equation (change line):

\begin{equation}
  a = b
\end{equation}

Change it to

\begin{equation}
  \begin{array}{l}
    a = \\ 
    b
  \end{array}
\end{equation}