Rwarm[a_, b_, \[Lambda]_,
  t_] := (1 - (1 - Exp[-(a^b)*(t^b)])^3)*(1 - (1 -
       Exp[-\[Lambda]*t])^2)
R[a_, b_, \[Rho]_, t_] := (1 - (1 - Exp[-(a^b)*(t^b)])^3)*
  Exp[-\[Rho]*t]
a = 1;
b = 1/2;
\[Lambda] = 1/2;
ExpectedRwarm = Integrate[Rwarm[a, b, \[Lambda], t], {t, 0, Infinity}];
N[%]
ExpectedRrho[\[Rho]_] :=
  Integrate[R[a, b, \[Rho], t], {t, 0, Infinity}];
d = FindRoot[ExpectedRwarm - ExpectedRrho[\[Rho]], {\[Rho], 1/2}]
Plot[Evaluate[ExpectedRrho[\[Rho]]], {\[Rho], 0, 1/2}]
R[a, b, \[Rho], t]
ExpectedRrho[\[Rho]]