Here is a self-contained sample how it should work: 
package test;
import javax.persistence.*;
import java.sql.Timestamp;
import java.time.LocalDateTime;
import java.util.List;
public class MyTestCase {
    public static void main(String[] args)  {
        EntityManagerFactory emf =
                Persistence.createEntityManagerFactory(
                        "objectdb:$objectdb/db/test.tmp;drop");
        EntityManager em = emf.createEntityManager();
        em.getTransaction().begin();
        MyEntity e = new MyEntity();
        e.setCreated(LocalDateTime.now());
        em.persist(e);
        em.getTransaction().commit();
        Query query = em.createQuery("SELECT e FROM MyEntity e");
        List<MyEntity> resultList = query.getResultList();
        System.out.println(resultList.get(0).getCreated());
        em.close();
        emf.close();
    }
    @Entity
    public static class MyEntity {
        @Id
        private Long id;
        @Convert(converter = LocalDateTimeConverter.class)
        private LocalDateTime created;
        protected MyEntity() {}
        public Long getId() { return id; }
        public void setId(Long id) { this.id = id; }
        public LocalDateTime getCreated() { return created; }
        public void setCreated(LocalDateTime created) { this.created = created; }
    }
    @Converter(autoApply = true)
    public static class LocalDateTimeConverter implements AttributeConverter<LocalDateTime, Timestamp> {
        @Override
        public java.sql.Timestamp convertToDatabaseColumn(java.time.LocalDateTime attribute) {
            return attribute == null ? null : java.sql.Timestamp.valueOf(attribute);
        }
        @Override
        public java.time.LocalDateTime convertToEntityAttribute(java.sql.Timestamp dbData) {
            return dbData == null ? null : dbData.toLocalDateTime();
        }
    }
} Of course, as ObjectDB doesn't recognize the @Convert annotation defined in the JPA standard, I get an rollback caused by this exception: 
com.objectdb.o.UserException: Attempt to store an instance of a non persistable type java.time.LocalDateTime - field test.MyTestCase$MyEntity.created