Why is thallium(III) oxide stable while thallium(III) chloride unstable
The stability of thallium(III) oxide and thallium(III) chloride can be compared by observing two properties:


Size of corresponding ions 


Size of $\ce{Tl^3+}$ is 0.95 Å, size of $\ce{O^2-}$ is 1.44  Å and that of $\ce{Cl-}$ is 1.8  Å. So, it pretty seems that difference in size between thallium and oxygen atoms is less than that of difference in size between thallium and chlorine atoms. So, thallium(III) oxide is stable than that of thallium(III) chloride. (data)


Difference in electronegativity


Electronegativity of thallium is 2.04, oxygen is 3.44 and that of chlorine is 3.16. So, difference in electronegativity of thallium-oxygen is more than that of thallium-chlorine. So, thallium(III) oxide is stable than that of thallium(III) chloride. (data)