For purposes of trademark law, "Cave" and "The Cave" are far too close for comfort, imo.
If I search for "Cave" on Google, I'll get the geological thing. If I search "cave game" instead - whoops, I get the Minecraft thing. And also "Cave, The".
There's also the game studio CAVE, known for their shmups of course.
So this is an absolute nightmare in terms of findability (deliberately avoiding the term "SEO" here - this isn't SEO, it's just avoiding a massive footgun).