The following is intended to expand on hyportnex's comment in one particular direction; you may or may not find it useful.
Your path from the conservation equation of an extensive quantity $X$ (I'll use $\rho$ for something else later)
$$\frac{\partial X}{\partial t}+ \vec\nabla\cdot\vec J_X=0\tag{1}$$
straight to a heat-equation-like relation where $X$ is replaced with the temperature $T$ arguably requires clarification, as the temperature is not extensive or conserved.
A better link might start with conservation of energy density $u$ in a control volume. Then, $du=\rho c\,dT$ (with density $\rho$ and heat capacity $c$), and if only conduction is considered, Fourier's law gives heat flux $\vec J_Q=-k\nabla T$, yielding
$$\frac{\partial T}{\partial t}- \alpha\nabla^2 T=0,\tag{2}$$
where the thermal diffusivity $\alpha=\frac{k}{c\rho}$, where material properties are assumed to be constant, and where $T=T(x,y,z,t)$. This is the conduction heat equation.
One could instead work with the even simpler
$$\frac{\partial T}{\partial t}- \nabla^2 T=0,\tag{3}$$
given the implicit acceptance that now $T=T\left(x,y,z,\alpha t\right)$ or $T=T\left(\frac{x}{\sqrt{\alpha}},\frac{y}{\sqrt{\alpha}},\frac{z}{\sqrt{\alpha}},t\right)$.
Returning to the conservation equation, we might ask: What is the extensive conjugate pair to temperature? It is entropy $S$ (or entropy density $s$ in a control volume). However, entropy isn't conserved but rather is paraconserved, so to speak; put another way, $dS\ge 0$ in an isolated system—the Second Law. The following is adapted from Balluffi, Allen, and Carter's Kinetics of Materials. We could write a paraconservation equation
$$\frac{\partial s}{\partial t}+ \vec\nabla\cdot\vec J_S=\dot\sigma,\tag{4}$$
where $\dot\sigma\ge 0$ is the entropy density generation rate. From the time derivative of the First Law (considering heat transfer only, so $du=T\,ds$, and with minimal temperature changes) and then applying the energy conservation equation,
$$\frac{du}{dt}=T\frac{ds}{dt};$$
$$\frac{\partial s}{\partial t}=\frac{1}{T}\left(\frac{\partial u}{\partial t}\right)=-\frac{1}{T}\left(\vec\nabla\cdot\vec J_Q\right),$$
and because
$$-\frac{1}{T}\left(\vec\nabla\cdot \vec J_Q\right)=-\vec\nabla\cdot\left(\frac{\vec J_Q}{T}\right)+\nabla\left(\frac{1}{T}\right)\cdot \vec J_Q=-\vec\nabla\cdot\left(\frac{-k\nabla T}{T}\right)+\frac{k}{T^2}\left(\nabla T\right)^2,$$
we have
$$\frac{\partial s}{\partial t}+\vec\nabla\cdot\left(\frac{-k\nabla T}{T}\right)=\frac{k}{T^2}\left(\nabla T\right)^2.\tag{5}$$
Matching the terms to (4), we associate the entropy flux $\vec J_S$ with $\frac{-k\nabla T}{T}$ and estimate the entropy density generation rate as
$$\dot\sigma=\frac{k}{T^2}\left(\nabla T\right)^2;$$
thus, this rate scales up with the square of the temperature gradient down which heat is flowing. In this way, additionally, the First and Second Laws prohibit a negative thermal conductivity $k$. (5) provides the paraconservation equation for heat transfer.
Thus, although I don't agree that moving from (1) to (3) is as simple as replacing conserved quantity $X$ with temperature $T$, one can certainly apply the (para)conservation equations (1) and (4) to a variety of systems. In the case of diffusion, for example, matter is conserved, and the entropy density generation rate scales up with the square of the chemical potential gradient (in ideal systems, the square of the concentration gradient), with the key material property—in this case, the mobility—again always positive. Or one could apply the same treatment to (conserved) charge, for instance, and confirm an entropy generation rate of $\frac{V^2}{RT}$ at temperature $T$, consistent with the well-known power dissipation $I^2R$ of resistive heating.
