What you are seeing is the boundary of the parametric surface, at theta=0 and theta=Pi. To get rid of the boundary use BoundaryStyle:
With[{x = Sin[\[Phi]]*Cos[\[Theta]], y = Sin[\[Phi]]*Sin[\[Theta]],
z = Cos[\[Phi]]},
SphericalPlot3D[
5*x^2 + 10*y^2 + 3*z^2, {\[Phi], 0, 2 \[Pi]}, {\[Theta], 0, \[Pi]},
ColorFunction -> (ColorData["TemperatureMap"][#6] &),
MeshFunctions -> {#6 &}, Mesh -> {8}, BoundaryStyle -> None]]
