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}
