S=1/2;
J1=10;
T1= 0.1;

ω[x_,y_,z_,S1_]:=6 J1 S1 (Sqrt[1-(1/9) (((Cos[x]+Cos[y]+Cos[ z]))^2)])

Φ[S1_,T_]:=(1/(π^3))NIntegrate[(1/(Sqrt[1-(1/9) (((Cos[x]+Cos[y]+Cos[ z]))^2)]))Coth[((ω[x,y,z,S1])/(2 T))],{x,0,π},{y,0,π},{z,0,π}]

σ[S1_,T_]:=(1/2)(1/(Φ[S1,T]))

(1/2)((((1/(π^3))NIntegrate[(1/(Sqrt[1-(1/9) (((Cos[x]+Cos[y]+Cos[ z]))^2)])),{x,0,π},{y,0,π},{z,0,π}]))^(-1))

σ[0.45,1]