I had new wooden gates added on to my house. Put Thompson Water seal on the new wood gate. Can I now use linseed oil over top to help protect it from the sun and heat of Phoenix AZ???
NO. Linseed oil will not work on freshly sealed wood, it will not penetrate and will just lay oily on the surface until it partially dries into a gooey mess.
The benefit of products like Thompson's is that they last longer than natural oil finishes (although any claims of better protection are unproven, and in my opinion dubious). If you prefer linseed oil then you will have to wait a few (3?) years, then be prepared to re-apply the linseed oil annually. Better to just reapply the Thompsons every 2-3 years IMO.