Is there any investment model with time-varying depreciation rate? Is there investment model with different depreciation rate. 

For example, d1(depreciation rate in 1 period), d2(depreciation rate in 2 period)
A type of model you might be interested on is one where the depreciation rate is endogenous. This means, it depends on the value of other variables in the model, instead of being exogenous and constant (naturally, you can have a variable depreciation rate but exogenous, but that is as ad-hoc as a constant depreciation rate).

One example is this model. Equation 19.4 shows the formula of the depreciaton rate (which the authors also coll net utilization rate). Then, they state:


  Thus the adjustment path for the capital utilization rate is a flexible accelerator.
  This result rigorously establishes the empirical finding obtained by Nadiri and Rosen (1969) that the adjustment' path of the
  capital utilization rate is similar to the paths of capital and labor growth rates which are governed by flexible accelerators. Along the
  dynamic path, as the capital intensity decreases the net utilization rate declines for two reasons. First, there is the direct effect of the
  capital intensity on the net utilization rate. A decrease in the capital intensity leads the firm to reallocate resources towards capital output
  which decreases the rate. Second, there is the indirect effect, which arises because the decrease in capital intensity causes the price of
  unutilized capital to increase. Since the marginal value of unutilized capital increases, the firm then utilizes less of its capital and so the
  utilization rate falls. Thus $k^° &gt; k^*$ implies that $\delta^° &gt; \delta^*$. These results mean that along the dynamic path the net capital utilization rate
  and the rate of capital investment are inversely related while the net capital utilization rate and the labor investment rate are directly
  correlated.


This paper presents another model. They show that "embodied" technical change (i.e. technical change which comes with new machines rather than occur amid current machines) decreases the lifetime of capital, thereby increasing the depreciation rate.

This paper finds empirical support for a variable depreciation rate.