Can a sympathetic Gentile worship in synagogue?
Answer:
It is permitted for a Gentile to enter a synagogue, and for him to to pray there to God.
Sources: Referring to the Temple, the verse (Isa. 57:7) states that “My House is a house of prayer for all the nations.” Although it is not exactly common practice, there is nothing forbidden about a non-Jew entering a synagogue for the purpose of prayer to God. A further source for this is the prohibition on non-Jews to enter certain parts of the Temple (see Keilim 1:8), implying that the other parts of the sanctified area is open to entry of non-Jews.