It depends on which density you want: bulk density or density of the solid fraction only, and of course how precisely you want to determine the value.

With solids or liquids, density is easy to determine, all you need to do is establish the mass (weight), which you use with a scale, and volume, which you can typically get by adding the material to a known volume of liquid in a measuring device like a graduated cylinder, and measure the volume change after addition. You could actually do this at the same time, by having the volumetric container on a scale when you add the liquid or solid. Get both volume and weight change in one move.

This will work with a sponge too, to get the density of the sponge material (assuming that you have total permeability, no blocked void spaces when immersed in liquid).

If you want a bulk density of something porous like a sponge, the best method is to measure mass (weight) dry and wet (saturated). The volume of the open space is defined by the mass of water contained within that open space (the change in mass from dry to wet is due to contained water only, and we have established water density to more precision than you will likely need for your purposes-you probably won't need to correct for temperature).

I don't do gas measurements, which generally has a pressure dependence and you would need to do a pressure-mass curve using a fixed volume container (a glass bulb of known volume, seems a good way to go). Empty the glass bulb of all air, measure its weight, then inject it with the target gas, and measure its weight. The change will be due solely to the added gas mass. Do that at a few different pressures to establish the pressure-density curve.

You could, conceivably, crush the porous solid to eliminate pore space and then get the powder mass and volume the same as with any solid. From that, you could establish the pore volume in the original sample (measure volume of the uncrushed sample and its weight and go from there with some simple math).

If you want to determine surface area of a porous solid, well, then you would need to go into more complicated or higher tech methods.