So looking at an old physics textbook you can work out the gravitational acceleration on the surface of the Earth as g = GM/R^2, where the mass of the earth is M, radius R and G is your gravitational constant. The density of a sphere (I'll call this D as I can't get greek stuff to work here) (ignoring the oblate spheroid/geoid shape of a real planet) D = M/V and V = 4/3 Pi R^3.
If you combine those two equations you get g = G * D * 4/3 * Pi * R,
So yeah, you do have to double the density since g is nicely linear in R and D, meaning if you halve R you'll have to double D to get the same g.
That's the average density over the whole planet, to get this kind of extra mass you could have a metal rich structure, maybe a little like Mercury See here. You'd need a very large metallic core relative to mantle size, maybe 150-180% the size of the one on earth, and an exceptionally thin crust.
The planets core, given it's small size will likely be solid as it will have cooled quite quickly after formation, so there won't be much of a magnetic field, so you'd get a lot of nasty solar radiation on it's surface if it's close enough to it's parent body to be tropical.
If you REALLY wanted it shielded by radiation you'd need more fissile material in it's core, which would mean it would have to form in an environment rich in supernova remnants, as that's likely the only place elements larger than iron can form. So the sky might be quite colourful at night.
You'd also have an interesting birth for this sort of planet, Mercury is thought to be the remnant of a larger planet whose surface and most if it's mantle was removed in a catastrophic collision early in it's history (see this post).